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Theorem connpconn 33829
Description: A connected and locally path-connected space is path-connected. (Contributed by Mario Carneiro, 7-Jul-2015.)
Assertion
Ref Expression
connpconn ((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) β†’ 𝐽 ∈ PConn)

Proof of Theorem connpconn
Dummy variables π‘₯ 𝑓 𝑦 𝑧 𝑔 β„Ž 𝑠 𝑒 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 conntop 22768 . . 3 (𝐽 ∈ Conn β†’ 𝐽 ∈ Top)
21adantr 481 . 2 ((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) β†’ 𝐽 ∈ Top)
3 eqid 2736 . . . . . 6 βˆͺ 𝐽 = βˆͺ 𝐽
4 simpll 765 . . . . . 6 (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ 𝐽 ∈ Conn)
5 inss1 4188 . . . . . . 7 (𝐽 ∩ (Clsdβ€˜π½)) βŠ† 𝐽
6 simplr 767 . . . . . . . . . . . 12 (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ 𝐽 ∈ 𝑛-Locally PConn)
71ad2antrr 724 . . . . . . . . . . . . 13 (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ 𝐽 ∈ Top)
83topopn 22255 . . . . . . . . . . . . 13 (𝐽 ∈ Top β†’ βˆͺ 𝐽 ∈ 𝐽)
97, 8syl 17 . . . . . . . . . . . 12 (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ βˆͺ 𝐽 ∈ 𝐽)
10 simprr 771 . . . . . . . . . . . 12 (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ 𝑧 ∈ βˆͺ 𝐽)
11 nlly2i 22827 . . . . . . . . . . . 12 ((𝐽 ∈ 𝑛-Locally PConn ∧ βˆͺ 𝐽 ∈ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽) β†’ βˆƒπ‘  ∈ 𝒫 βˆͺ π½βˆƒπ‘’ ∈ 𝐽 (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))
126, 9, 10, 11syl3anc 1371 . . . . . . . . . . 11 (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ βˆƒπ‘  ∈ 𝒫 βˆͺ π½βˆƒπ‘’ ∈ 𝐽 (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))
13 simprr1 1221 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) β†’ 𝑧 ∈ 𝑒)
14 eqeq2 2748 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑀 β†’ ((π‘“β€˜1) = 𝑦 ↔ (π‘“β€˜1) = 𝑀))
1514anbi2d 629 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑀 β†’ (((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦) ↔ ((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑀)))
1615rexbidv 3175 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑀 β†’ (βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦) ↔ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑀)))
1716elrab 3645 . . . . . . . . . . . . . . . . . 18 (𝑀 ∈ {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)} ↔ (𝑀 ∈ βˆͺ 𝐽 ∧ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑀)))
1817simprbi 497 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)} β†’ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑀))
19 simprr3 1223 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) β†’ (𝐽 β†Ύt 𝑠) ∈ PConn)
2019adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) β†’ (𝐽 β†Ύt 𝑠) ∈ PConn)
21 simprr2 1222 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) β†’ 𝑒 βŠ† 𝑠)
2221adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) β†’ 𝑒 βŠ† 𝑠)
23 simprll 777 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) β†’ 𝑀 ∈ 𝑒)
2422, 23sseldd 3945 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) β†’ 𝑀 ∈ 𝑠)
257ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) β†’ 𝐽 ∈ Top)
26 elpwi 4567 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑠 ∈ 𝒫 βˆͺ 𝐽 β†’ 𝑠 βŠ† βˆͺ 𝐽)
2726ad2antrl 726 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) β†’ 𝑠 βŠ† βˆͺ 𝐽)
2827adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) β†’ 𝑠 βŠ† βˆͺ 𝐽)
293restuni 22513 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐽 ∈ Top ∧ 𝑠 βŠ† βˆͺ 𝐽) β†’ 𝑠 = βˆͺ (𝐽 β†Ύt 𝑠))
3025, 28, 29syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) β†’ 𝑠 = βˆͺ (𝐽 β†Ύt 𝑠))
3124, 30eleqtrd 2839 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) β†’ 𝑀 ∈ βˆͺ (𝐽 β†Ύt 𝑠))
32 simprr 771 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) β†’ 𝑦 ∈ 𝑒)
3322, 32sseldd 3945 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) β†’ 𝑦 ∈ 𝑠)
3433, 30eleqtrd 2839 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) β†’ 𝑦 ∈ βˆͺ (𝐽 β†Ύt 𝑠))
35 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . . . . 26 βˆͺ (𝐽 β†Ύt 𝑠) = βˆͺ (𝐽 β†Ύt 𝑠)
3635pconncn 33818 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐽 β†Ύt 𝑠) ∈ PConn ∧ 𝑀 ∈ βˆͺ (𝐽 β†Ύt 𝑠) ∧ 𝑦 ∈ βˆͺ (𝐽 β†Ύt 𝑠)) β†’ βˆƒβ„Ž ∈ (II Cn (𝐽 β†Ύt 𝑠))((β„Žβ€˜0) = 𝑀 ∧ (β„Žβ€˜1) = 𝑦))
3720, 31, 34, 36syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) β†’ βˆƒβ„Ž ∈ (II Cn (𝐽 β†Ύt 𝑠))((β„Žβ€˜0) = 𝑀 ∧ (β„Žβ€˜1) = 𝑦))
38 simplrl 775 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒) β†’ 𝑔 ∈ (II Cn 𝐽))
3938ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) ∧ (β„Ž ∈ (II Cn (𝐽 β†Ύt 𝑠)) ∧ ((β„Žβ€˜0) = 𝑀 ∧ (β„Žβ€˜1) = 𝑦))) β†’ 𝑔 ∈ (II Cn 𝐽))
4025adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) ∧ (β„Ž ∈ (II Cn (𝐽 β†Ύt 𝑠)) ∧ ((β„Žβ€˜0) = 𝑀 ∧ (β„Žβ€˜1) = 𝑦))) β†’ 𝐽 ∈ Top)
41 cnrest2r 22638 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐽 ∈ Top β†’ (II Cn (𝐽 β†Ύt 𝑠)) βŠ† (II Cn 𝐽))
4240, 41syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) ∧ (β„Ž ∈ (II Cn (𝐽 β†Ύt 𝑠)) ∧ ((β„Žβ€˜0) = 𝑀 ∧ (β„Žβ€˜1) = 𝑦))) β†’ (II Cn (𝐽 β†Ύt 𝑠)) βŠ† (II Cn 𝐽))
43 simprl 769 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) ∧ (β„Ž ∈ (II Cn (𝐽 β†Ύt 𝑠)) ∧ ((β„Žβ€˜0) = 𝑀 ∧ (β„Žβ€˜1) = 𝑦))) β†’ β„Ž ∈ (II Cn (𝐽 β†Ύt 𝑠)))
4442, 43sseldd 3945 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) ∧ (β„Ž ∈ (II Cn (𝐽 β†Ύt 𝑠)) ∧ ((β„Žβ€˜0) = 𝑀 ∧ (β„Žβ€˜1) = 𝑦))) β†’ β„Ž ∈ (II Cn 𝐽))
45 simplrr 776 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒) β†’ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))
4645ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) ∧ (β„Ž ∈ (II Cn (𝐽 β†Ύt 𝑠)) ∧ ((β„Žβ€˜0) = 𝑀 ∧ (β„Žβ€˜1) = 𝑦))) β†’ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))
4746simprd 496 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) ∧ (β„Ž ∈ (II Cn (𝐽 β†Ύt 𝑠)) ∧ ((β„Žβ€˜0) = 𝑀 ∧ (β„Žβ€˜1) = 𝑦))) β†’ (π‘”β€˜1) = 𝑀)
48 simprrl 779 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) ∧ (β„Ž ∈ (II Cn (𝐽 β†Ύt 𝑠)) ∧ ((β„Žβ€˜0) = 𝑀 ∧ (β„Žβ€˜1) = 𝑦))) β†’ (β„Žβ€˜0) = 𝑀)
4947, 48eqtr4d 2779 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) ∧ (β„Ž ∈ (II Cn (𝐽 β†Ύt 𝑠)) ∧ ((β„Žβ€˜0) = 𝑀 ∧ (β„Žβ€˜1) = 𝑦))) β†’ (π‘”β€˜1) = (β„Žβ€˜0))
5039, 44, 49pcocn 24380 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) ∧ (β„Ž ∈ (II Cn (𝐽 β†Ύt 𝑠)) ∧ ((β„Žβ€˜0) = 𝑀 ∧ (β„Žβ€˜1) = 𝑦))) β†’ (𝑔(*π‘β€˜π½)β„Ž) ∈ (II Cn 𝐽))
5139, 44pco0 24377 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) ∧ (β„Ž ∈ (II Cn (𝐽 β†Ύt 𝑠)) ∧ ((β„Žβ€˜0) = 𝑀 ∧ (β„Žβ€˜1) = 𝑦))) β†’ ((𝑔(*π‘β€˜π½)β„Ž)β€˜0) = (π‘”β€˜0))
5246simpld 495 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) ∧ (β„Ž ∈ (II Cn (𝐽 β†Ύt 𝑠)) ∧ ((β„Žβ€˜0) = 𝑀 ∧ (β„Žβ€˜1) = 𝑦))) β†’ (π‘”β€˜0) = π‘₯)
5351, 52eqtrd 2776 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) ∧ (β„Ž ∈ (II Cn (𝐽 β†Ύt 𝑠)) ∧ ((β„Žβ€˜0) = 𝑀 ∧ (β„Žβ€˜1) = 𝑦))) β†’ ((𝑔(*π‘β€˜π½)β„Ž)β€˜0) = π‘₯)
5439, 44pco1 24378 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) ∧ (β„Ž ∈ (II Cn (𝐽 β†Ύt 𝑠)) ∧ ((β„Žβ€˜0) = 𝑀 ∧ (β„Žβ€˜1) = 𝑦))) β†’ ((𝑔(*π‘β€˜π½)β„Ž)β€˜1) = (β„Žβ€˜1))
55 simprrr 780 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) ∧ (β„Ž ∈ (II Cn (𝐽 β†Ύt 𝑠)) ∧ ((β„Žβ€˜0) = 𝑀 ∧ (β„Žβ€˜1) = 𝑦))) β†’ (β„Žβ€˜1) = 𝑦)
5654, 55eqtrd 2776 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) ∧ (β„Ž ∈ (II Cn (𝐽 β†Ύt 𝑠)) ∧ ((β„Žβ€˜0) = 𝑀 ∧ (β„Žβ€˜1) = 𝑦))) β†’ ((𝑔(*π‘β€˜π½)β„Ž)β€˜1) = 𝑦)
57 fveq1 6841 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑓 = (𝑔(*π‘β€˜π½)β„Ž) β†’ (π‘“β€˜0) = ((𝑔(*π‘β€˜π½)β„Ž)β€˜0))
5857eqeq1d 2738 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓 = (𝑔(*π‘β€˜π½)β„Ž) β†’ ((π‘“β€˜0) = π‘₯ ↔ ((𝑔(*π‘β€˜π½)β„Ž)β€˜0) = π‘₯))
59 fveq1 6841 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑓 = (𝑔(*π‘β€˜π½)β„Ž) β†’ (π‘“β€˜1) = ((𝑔(*π‘β€˜π½)β„Ž)β€˜1))
6059eqeq1d 2738 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓 = (𝑔(*π‘β€˜π½)β„Ž) β†’ ((π‘“β€˜1) = 𝑦 ↔ ((𝑔(*π‘β€˜π½)β„Ž)β€˜1) = 𝑦))
6158, 60anbi12d 631 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 = (𝑔(*π‘β€˜π½)β„Ž) β†’ (((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦) ↔ (((𝑔(*π‘β€˜π½)β„Ž)β€˜0) = π‘₯ ∧ ((𝑔(*π‘β€˜π½)β„Ž)β€˜1) = 𝑦)))
6261rspcev 3581 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑔(*π‘β€˜π½)β„Ž) ∈ (II Cn 𝐽) ∧ (((𝑔(*π‘β€˜π½)β„Ž)β€˜0) = π‘₯ ∧ ((𝑔(*π‘β€˜π½)β„Ž)β€˜1) = 𝑦)) β†’ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦))
6350, 53, 56, 62syl12anc 835 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) ∧ (β„Ž ∈ (II Cn (𝐽 β†Ύt 𝑠)) ∧ ((β„Žβ€˜0) = 𝑀 ∧ (β„Žβ€˜1) = 𝑦))) β†’ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦))
6437, 63rexlimddv 3158 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ ((𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) ∧ 𝑦 ∈ 𝑒)) β†’ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦))
6564anassrs 468 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ (𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀)))) ∧ 𝑦 ∈ 𝑒) β†’ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦))
6665ralrimiva 3143 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ (𝑀 ∈ 𝑒 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀)))) β†’ βˆ€π‘¦ ∈ 𝑒 βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦))
6766anassrs 468 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ 𝑀 ∈ 𝑒) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))) β†’ βˆ€π‘¦ ∈ 𝑒 βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦))
6867rexlimdvaa 3153 . . . . . . . . . . . . . . . . . . 19 (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ 𝑀 ∈ 𝑒) β†’ (βˆƒπ‘” ∈ (II Cn 𝐽)((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀) β†’ βˆ€π‘¦ ∈ 𝑒 βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)))
6921adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ 𝑀 ∈ 𝑒) β†’ 𝑒 βŠ† 𝑠)
70 simplrl 775 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ 𝑀 ∈ 𝑒) β†’ 𝑠 ∈ 𝒫 βˆͺ 𝐽)
7170, 26syl 17 . . . . . . . . . . . . . . . . . . . 20 (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ 𝑀 ∈ 𝑒) β†’ 𝑠 βŠ† βˆͺ 𝐽)
7269, 71sstrd 3954 . . . . . . . . . . . . . . . . . . 19 (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ 𝑀 ∈ 𝑒) β†’ 𝑒 βŠ† βˆͺ 𝐽)
7368, 72jctild 526 . . . . . . . . . . . . . . . . . 18 (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ 𝑀 ∈ 𝑒) β†’ (βˆƒπ‘” ∈ (II Cn 𝐽)((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀) β†’ (𝑒 βŠ† βˆͺ 𝐽 ∧ βˆ€π‘¦ ∈ 𝑒 βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦))))
74 fveq1 6841 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝑔 β†’ (π‘“β€˜0) = (π‘”β€˜0))
7574eqeq1d 2738 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝑔 β†’ ((π‘“β€˜0) = π‘₯ ↔ (π‘”β€˜0) = π‘₯))
76 fveq1 6841 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝑔 β†’ (π‘“β€˜1) = (π‘”β€˜1))
7776eqeq1d 2738 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝑔 β†’ ((π‘“β€˜1) = 𝑀 ↔ (π‘”β€˜1) = 𝑀))
7875, 77anbi12d 631 . . . . . . . . . . . . . . . . . . 19 (𝑓 = 𝑔 β†’ (((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑀) ↔ ((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀)))
7978cbvrexvw 3226 . . . . . . . . . . . . . . . . . 18 (βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑀) ↔ βˆƒπ‘” ∈ (II Cn 𝐽)((π‘”β€˜0) = π‘₯ ∧ (π‘”β€˜1) = 𝑀))
80 ssrab 4030 . . . . . . . . . . . . . . . . . 18 (𝑒 βŠ† {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)} ↔ (𝑒 βŠ† βˆͺ 𝐽 ∧ βˆ€π‘¦ ∈ 𝑒 βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)))
8173, 79, 803imtr4g 295 . . . . . . . . . . . . . . . . 17 (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ 𝑀 ∈ 𝑒) β†’ (βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑀) β†’ 𝑒 βŠ† {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)}))
8218, 81syl5 34 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) ∧ 𝑀 ∈ 𝑒) β†’ (𝑀 ∈ {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)} β†’ 𝑒 βŠ† {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)}))
8382ralrimiva 3143 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) β†’ βˆ€π‘€ ∈ 𝑒 (𝑀 ∈ {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)} β†’ 𝑒 βŠ† {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)}))
8413, 83jca 512 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ (𝑠 ∈ 𝒫 βˆͺ 𝐽 ∧ (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn))) β†’ (𝑧 ∈ 𝑒 ∧ βˆ€π‘€ ∈ 𝑒 (𝑀 ∈ {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)} β†’ 𝑒 βŠ† {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)})))
8584expr 457 . . . . . . . . . . . . 13 ((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ 𝑠 ∈ 𝒫 βˆͺ 𝐽) β†’ ((𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn) β†’ (𝑧 ∈ 𝑒 ∧ βˆ€π‘€ ∈ 𝑒 (𝑀 ∈ {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)} β†’ 𝑒 βŠ† {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)}))))
8685reximdv 3167 . . . . . . . . . . . 12 ((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) ∧ 𝑠 ∈ 𝒫 βˆͺ 𝐽) β†’ (βˆƒπ‘’ ∈ 𝐽 (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn) β†’ βˆƒπ‘’ ∈ 𝐽 (𝑧 ∈ 𝑒 ∧ βˆ€π‘€ ∈ 𝑒 (𝑀 ∈ {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)} β†’ 𝑒 βŠ† {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)}))))
8786rexlimdva 3152 . . . . . . . . . . 11 (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ (βˆƒπ‘  ∈ 𝒫 βˆͺ π½βˆƒπ‘’ ∈ 𝐽 (𝑧 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ PConn) β†’ βˆƒπ‘’ ∈ 𝐽 (𝑧 ∈ 𝑒 ∧ βˆ€π‘€ ∈ 𝑒 (𝑀 ∈ {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)} β†’ 𝑒 βŠ† {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)}))))
8812, 87mpd 15 . . . . . . . . . 10 (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (π‘₯ ∈ βˆͺ 𝐽 ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ βˆƒπ‘’ ∈ 𝐽 (𝑧 ∈ 𝑒 ∧ βˆ€π‘€ ∈ 𝑒 (𝑀 ∈ {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)} β†’ 𝑒 βŠ† {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)})))
8988anassrs 468 . . . . . . . . 9 ((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ π‘₯ ∈ βˆͺ 𝐽) ∧ 𝑧 ∈ βˆͺ 𝐽) β†’ βˆƒπ‘’ ∈ 𝐽 (𝑧 ∈ 𝑒 ∧ βˆ€π‘€ ∈ 𝑒 (𝑀 ∈ {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)} β†’ 𝑒 βŠ† {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)})))
9089ralrimiva 3143 . . . . . . . 8 (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ βˆ€π‘§ ∈ βˆͺ π½βˆƒπ‘’ ∈ 𝐽 (𝑧 ∈ 𝑒 ∧ βˆ€π‘€ ∈ 𝑒 (𝑀 ∈ {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)} β†’ 𝑒 βŠ† {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)})))
911ad2antrr 724 . . . . . . . . 9 (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ 𝐽 ∈ Top)
92 ssrab2 4037 . . . . . . . . 9 {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)} βŠ† βˆͺ 𝐽
933isclo2 22439 . . . . . . . . 9 ((𝐽 ∈ Top ∧ {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)} βŠ† βˆͺ 𝐽) β†’ ({𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)} ∈ (𝐽 ∩ (Clsdβ€˜π½)) ↔ βˆ€π‘§ ∈ βˆͺ π½βˆƒπ‘’ ∈ 𝐽 (𝑧 ∈ 𝑒 ∧ βˆ€π‘€ ∈ 𝑒 (𝑀 ∈ {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)} β†’ 𝑒 βŠ† {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)}))))
9491, 92, 93sylancl 586 . . . . . . . 8 (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ ({𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)} ∈ (𝐽 ∩ (Clsdβ€˜π½)) ↔ βˆ€π‘§ ∈ βˆͺ π½βˆƒπ‘’ ∈ 𝐽 (𝑧 ∈ 𝑒 ∧ βˆ€π‘€ ∈ 𝑒 (𝑀 ∈ {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)} β†’ 𝑒 βŠ† {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)}))))
9590, 94mpbird 256 . . . . . . 7 (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)} ∈ (𝐽 ∩ (Clsdβ€˜π½)))
965, 95sselid 3942 . . . . . 6 (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)} ∈ 𝐽)
97 simpr 485 . . . . . . . 8 (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ π‘₯ ∈ βˆͺ 𝐽)
98 iitopon 24242 . . . . . . . . . 10 II ∈ (TopOnβ€˜(0[,]1))
9998a1i 11 . . . . . . . . 9 (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ II ∈ (TopOnβ€˜(0[,]1)))
1003toptopon 22266 . . . . . . . . . 10 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
10191, 100sylib 217 . . . . . . . . 9 (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
102 cnconst2 22634 . . . . . . . . 9 ((II ∈ (TopOnβ€˜(0[,]1)) ∧ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽) ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ ((0[,]1) Γ— {π‘₯}) ∈ (II Cn 𝐽))
10399, 101, 97, 102syl3anc 1371 . . . . . . . 8 (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ ((0[,]1) Γ— {π‘₯}) ∈ (II Cn 𝐽))
104 0elunit 13386 . . . . . . . . 9 0 ∈ (0[,]1)
105 vex 3449 . . . . . . . . . 10 π‘₯ ∈ V
106105fvconst2 7153 . . . . . . . . 9 (0 ∈ (0[,]1) β†’ (((0[,]1) Γ— {π‘₯})β€˜0) = π‘₯)
107104, 106mp1i 13 . . . . . . . 8 (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ (((0[,]1) Γ— {π‘₯})β€˜0) = π‘₯)
108 1elunit 13387 . . . . . . . . 9 1 ∈ (0[,]1)
109105fvconst2 7153 . . . . . . . . 9 (1 ∈ (0[,]1) β†’ (((0[,]1) Γ— {π‘₯})β€˜1) = π‘₯)
110108, 109mp1i 13 . . . . . . . 8 (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ (((0[,]1) Γ— {π‘₯})β€˜1) = π‘₯)
111 eqeq2 2748 . . . . . . . . . 10 (𝑦 = π‘₯ β†’ ((π‘“β€˜1) = 𝑦 ↔ (π‘“β€˜1) = π‘₯))
112111anbi2d 629 . . . . . . . . 9 (𝑦 = π‘₯ β†’ (((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦) ↔ ((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = π‘₯)))
113 fveq1 6841 . . . . . . . . . . 11 (𝑓 = ((0[,]1) Γ— {π‘₯}) β†’ (π‘“β€˜0) = (((0[,]1) Γ— {π‘₯})β€˜0))
114113eqeq1d 2738 . . . . . . . . . 10 (𝑓 = ((0[,]1) Γ— {π‘₯}) β†’ ((π‘“β€˜0) = π‘₯ ↔ (((0[,]1) Γ— {π‘₯})β€˜0) = π‘₯))
115 fveq1 6841 . . . . . . . . . . 11 (𝑓 = ((0[,]1) Γ— {π‘₯}) β†’ (π‘“β€˜1) = (((0[,]1) Γ— {π‘₯})β€˜1))
116115eqeq1d 2738 . . . . . . . . . 10 (𝑓 = ((0[,]1) Γ— {π‘₯}) β†’ ((π‘“β€˜1) = π‘₯ ↔ (((0[,]1) Γ— {π‘₯})β€˜1) = π‘₯))
117114, 116anbi12d 631 . . . . . . . . 9 (𝑓 = ((0[,]1) Γ— {π‘₯}) β†’ (((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = π‘₯) ↔ ((((0[,]1) Γ— {π‘₯})β€˜0) = π‘₯ ∧ (((0[,]1) Γ— {π‘₯})β€˜1) = π‘₯)))
118112, 117rspc2ev 3592 . . . . . . . 8 ((π‘₯ ∈ βˆͺ 𝐽 ∧ ((0[,]1) Γ— {π‘₯}) ∈ (II Cn 𝐽) ∧ ((((0[,]1) Γ— {π‘₯})β€˜0) = π‘₯ ∧ (((0[,]1) Γ— {π‘₯})β€˜1) = π‘₯)) β†’ βˆƒπ‘¦ ∈ βˆͺ π½βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦))
11997, 103, 107, 110, 118syl112anc 1374 . . . . . . 7 (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ βˆƒπ‘¦ ∈ βˆͺ π½βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦))
120 rabn0 4345 . . . . . . 7 ({𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)} β‰  βˆ… ↔ βˆƒπ‘¦ ∈ βˆͺ π½βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦))
121119, 120sylibr 233 . . . . . 6 (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)} β‰  βˆ…)
122 inss2 4189 . . . . . . 7 (𝐽 ∩ (Clsdβ€˜π½)) βŠ† (Clsdβ€˜π½)
123122, 95sselid 3942 . . . . . 6 (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)} ∈ (Clsdβ€˜π½))
1243, 4, 96, 121, 123connclo 22766 . . . . 5 (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)} = βˆͺ 𝐽)
125124eqcomd 2742 . . . 4 (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ βˆͺ 𝐽 = {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)})
126 rabid2 3436 . . . 4 (βˆͺ 𝐽 = {𝑦 ∈ βˆͺ 𝐽 ∣ βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)} ↔ βˆ€π‘¦ ∈ βˆͺ π½βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦))
127125, 126sylib 217 . . 3 (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ βˆ€π‘¦ ∈ βˆͺ π½βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦))
128127ralrimiva 3143 . 2 ((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) β†’ βˆ€π‘₯ ∈ βˆͺ π½βˆ€π‘¦ ∈ βˆͺ π½βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦))
1293ispconn 33817 . 2 (𝐽 ∈ PConn ↔ (𝐽 ∈ Top ∧ βˆ€π‘₯ ∈ βˆͺ π½βˆ€π‘¦ ∈ βˆͺ π½βˆƒπ‘“ ∈ (II Cn 𝐽)((π‘“β€˜0) = π‘₯ ∧ (π‘“β€˜1) = 𝑦)))
1302, 128, 129sylanbrc 583 1 ((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) β†’ 𝐽 ∈ PConn)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2943  βˆ€wral 3064  βˆƒwrex 3073  {crab 3407   ∩ cin 3909   βŠ† wss 3910  βˆ…c0 4282  π’« cpw 4560  {csn 4586  βˆͺ cuni 4865   Γ— cxp 5631  β€˜cfv 6496  (class class class)co 7357  0cc0 11051  1c1 11052  [,]cicc 13267   β†Ύt crest 17302  Topctop 22242  TopOnctopon 22259  Clsdccld 22367   Cn ccn 22575  Conncconn 22762  π‘›-Locally cnlly 22816  IIcii 24238  *𝑝cpco 24363  PConncpconn 33813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-icc 13271  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-nei 22449  df-cn 22578  df-cnp 22579  df-conn 22763  df-nlly 22818  df-tx 22913  df-hmeo 23106  df-xms 23673  df-ms 23674  df-tms 23675  df-ii 24240  df-pco 24368  df-pconn 33815
This theorem is referenced by: (None)
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