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Theorem cvmlift3lem7 34991
Description: Lemma for cvmlift3 34994. (Contributed by Mario Carneiro, 9-Jul-2015.)
Hypotheses
Ref Expression
cvmlift3.b 𝐡 = βˆͺ 𝐢
cvmlift3.y π‘Œ = βˆͺ 𝐾
cvmlift3.f (πœ‘ β†’ 𝐹 ∈ (𝐢 CovMap 𝐽))
cvmlift3.k (πœ‘ β†’ 𝐾 ∈ SConn)
cvmlift3.l (πœ‘ β†’ 𝐾 ∈ 𝑛-Locally PConn)
cvmlift3.o (πœ‘ β†’ 𝑂 ∈ π‘Œ)
cvmlift3.g (πœ‘ β†’ 𝐺 ∈ (𝐾 Cn 𝐽))
cvmlift3.p (πœ‘ β†’ 𝑃 ∈ 𝐡)
cvmlift3.e (πœ‘ β†’ (πΉβ€˜π‘ƒ) = (πΊβ€˜π‘‚))
cvmlift3.h 𝐻 = (π‘₯ ∈ π‘Œ ↦ (℩𝑧 ∈ 𝐡 βˆƒπ‘“ ∈ (II Cn 𝐾)((π‘“β€˜0) = 𝑂 ∧ (π‘“β€˜1) = π‘₯ ∧ ((℩𝑔 ∈ (II Cn 𝐢)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (π‘”β€˜0) = 𝑃))β€˜1) = 𝑧)))
cvmlift3lem7.s 𝑆 = (π‘˜ ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐢 βˆ– {βˆ…}) ∣ (βˆͺ 𝑠 = (◑𝐹 β€œ π‘˜) ∧ βˆ€π‘ ∈ 𝑠 (βˆ€π‘‘ ∈ (𝑠 βˆ– {𝑐})(𝑐 ∩ 𝑑) = βˆ… ∧ (𝐹 β†Ύ 𝑐) ∈ ((𝐢 β†Ύt 𝑐)Homeo(𝐽 β†Ύt π‘˜))))})
cvmlift3lem7.1 (πœ‘ β†’ (πΊβ€˜π‘‹) ∈ 𝐴)
cvmlift3lem7.2 (πœ‘ β†’ 𝑇 ∈ (π‘†β€˜π΄))
cvmlift3lem7.3 (πœ‘ β†’ 𝑀 βŠ† (◑𝐺 β€œ 𝐴))
cvmlift3lem7.w π‘Š = (℩𝑏 ∈ 𝑇 (π»β€˜π‘‹) ∈ 𝑏)
cvmlift3lem7.7 (πœ‘ β†’ (𝐾 β†Ύt 𝑀) ∈ PConn)
cvmlift3lem7.4 (πœ‘ β†’ 𝑉 ∈ 𝐾)
cvmlift3lem7.5 (πœ‘ β†’ 𝑉 βŠ† 𝑀)
cvmlift3lem7.6 (πœ‘ β†’ 𝑋 ∈ 𝑉)
Assertion
Ref Expression
cvmlift3lem7 (πœ‘ β†’ 𝐻 ∈ ((𝐾 CnP 𝐢)β€˜π‘‹))
Distinct variable groups:   𝑏,𝑐,𝑑,𝑓,π‘˜,𝑠,𝑧,𝐴   𝑓,𝑔,𝑧,𝑏,π‘₯   𝐽,𝑏   𝑔,𝑐,π‘₯,𝐽,𝑑,𝑓,π‘˜,𝑠   𝐹,𝑏,𝑐,𝑑,𝑓,𝑔,π‘˜,𝑠   π‘₯,𝑧,𝐹   𝑓,𝑀,𝑔,π‘₯   𝐻,𝑏,𝑐,𝑑,𝑓,𝑔,π‘₯,𝑧   𝑆,𝑏,𝑓,π‘₯   𝐡,𝑏,𝑑,𝑓,𝑔,π‘₯,𝑧   𝑋,𝑏,𝑐,𝑑,𝑓,𝑔,π‘₯,𝑧   𝐺,𝑏,𝑐,𝑑,𝑓,𝑔,π‘˜,π‘₯,𝑧   𝑇,𝑏,𝑐,𝑑,𝑠   𝐢,𝑏,𝑐,𝑑,𝑓,𝑔,π‘˜,𝑠,π‘₯,𝑧   πœ‘,𝑓,π‘₯   𝐾,𝑏,𝑐,𝑓,𝑔,π‘₯,𝑧   𝑃,𝑏,𝑐,𝑑,𝑓,𝑔,π‘₯,𝑧   𝑂,𝑏,𝑐,𝑓,𝑔,π‘₯,𝑧   𝑓,π‘Œ,𝑔,π‘₯,𝑧   π‘Š,𝑐,𝑑,𝑓,π‘₯
Allowed substitution hints:   πœ‘(𝑧,𝑔,π‘˜,𝑠,𝑏,𝑐,𝑑)   𝐴(π‘₯,𝑔)   𝐡(π‘˜,𝑠,𝑐)   𝑃(π‘˜,𝑠)   𝑆(𝑧,𝑔,π‘˜,𝑠,𝑐,𝑑)   𝑇(π‘₯,𝑧,𝑓,𝑔,π‘˜)   𝐺(𝑠)   𝐻(π‘˜,𝑠)   𝐽(𝑧)   𝐾(π‘˜,𝑠,𝑑)   𝑀(𝑧,π‘˜,𝑠,𝑏,𝑐,𝑑)   𝑂(π‘˜,𝑠,𝑑)   𝑉(π‘₯,𝑧,𝑓,𝑔,π‘˜,𝑠,𝑏,𝑐,𝑑)   π‘Š(𝑧,𝑔,π‘˜,𝑠,𝑏)   𝑋(π‘˜,𝑠)   π‘Œ(π‘˜,𝑠,𝑏,𝑐,𝑑)

Proof of Theorem cvmlift3lem7
Dummy variables π‘Ž 𝑦 β„Ž 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cvmlift3.b . . . 4 𝐡 = βˆͺ 𝐢
2 cvmlift3.y . . . 4 π‘Œ = βˆͺ 𝐾
3 cvmlift3lem7.s . . . 4 𝑆 = (π‘˜ ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐢 βˆ– {βˆ…}) ∣ (βˆͺ 𝑠 = (◑𝐹 β€œ π‘˜) ∧ βˆ€π‘ ∈ 𝑠 (βˆ€π‘‘ ∈ (𝑠 βˆ– {𝑐})(𝑐 ∩ 𝑑) = βˆ… ∧ (𝐹 β†Ύ 𝑐) ∈ ((𝐢 β†Ύt 𝑐)Homeo(𝐽 β†Ύt π‘˜))))})
4 cvmlift3.f . . . 4 (πœ‘ β†’ 𝐹 ∈ (𝐢 CovMap 𝐽))
5 cvmlift3.k . . . . 5 (πœ‘ β†’ 𝐾 ∈ SConn)
6 cvmlift3.l . . . . 5 (πœ‘ β†’ 𝐾 ∈ 𝑛-Locally PConn)
7 cvmlift3.o . . . . 5 (πœ‘ β†’ 𝑂 ∈ π‘Œ)
8 cvmlift3.g . . . . 5 (πœ‘ β†’ 𝐺 ∈ (𝐾 Cn 𝐽))
9 cvmlift3.p . . . . 5 (πœ‘ β†’ 𝑃 ∈ 𝐡)
10 cvmlift3.e . . . . 5 (πœ‘ β†’ (πΉβ€˜π‘ƒ) = (πΊβ€˜π‘‚))
11 cvmlift3.h . . . . 5 𝐻 = (π‘₯ ∈ π‘Œ ↦ (℩𝑧 ∈ 𝐡 βˆƒπ‘“ ∈ (II Cn 𝐾)((π‘“β€˜0) = 𝑂 ∧ (π‘“β€˜1) = π‘₯ ∧ ((℩𝑔 ∈ (II Cn 𝐢)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (π‘”β€˜0) = 𝑃))β€˜1) = 𝑧)))
121, 2, 4, 5, 6, 7, 8, 9, 10, 11cvmlift3lem3 34987 . . . 4 (πœ‘ β†’ 𝐻:π‘ŒβŸΆπ΅)
131, 2, 4, 5, 6, 7, 8, 9, 10, 11cvmlift3lem5 34989 . . . . 5 (πœ‘ β†’ (𝐹 ∘ 𝐻) = 𝐺)
1413, 8eqeltrd 2825 . . . 4 (πœ‘ β†’ (𝐹 ∘ 𝐻) ∈ (𝐾 Cn 𝐽))
15 sconntop 34894 . . . . 5 (𝐾 ∈ SConn β†’ 𝐾 ∈ Top)
165, 15syl 17 . . . 4 (πœ‘ β†’ 𝐾 ∈ Top)
17 cvmlift3lem7.3 . . . . . 6 (πœ‘ β†’ 𝑀 βŠ† (◑𝐺 β€œ 𝐴))
18 cnvimass 6080 . . . . . . 7 (◑𝐺 β€œ 𝐴) βŠ† dom 𝐺
19 eqid 2725 . . . . . . . . 9 βˆͺ 𝐽 = βˆͺ 𝐽
202, 19cnf 23166 . . . . . . . 8 (𝐺 ∈ (𝐾 Cn 𝐽) β†’ 𝐺:π‘ŒβŸΆβˆͺ 𝐽)
21 fdm 6725 . . . . . . . 8 (𝐺:π‘ŒβŸΆβˆͺ 𝐽 β†’ dom 𝐺 = π‘Œ)
228, 20, 213syl 18 . . . . . . 7 (πœ‘ β†’ dom 𝐺 = π‘Œ)
2318, 22sseqtrid 4025 . . . . . 6 (πœ‘ β†’ (◑𝐺 β€œ 𝐴) βŠ† π‘Œ)
2417, 23sstrd 3983 . . . . 5 (πœ‘ β†’ 𝑀 βŠ† π‘Œ)
25 cvmlift3lem7.5 . . . . . 6 (πœ‘ β†’ 𝑉 βŠ† 𝑀)
26 cvmlift3lem7.6 . . . . . 6 (πœ‘ β†’ 𝑋 ∈ 𝑉)
2725, 26sseldd 3973 . . . . 5 (πœ‘ β†’ 𝑋 ∈ 𝑀)
2824, 27sseldd 3973 . . . 4 (πœ‘ β†’ 𝑋 ∈ π‘Œ)
29 cvmlift3lem7.2 . . . 4 (πœ‘ β†’ 𝑇 ∈ (π‘†β€˜π΄))
3012, 28ffvelcdmd 7089 . . . . 5 (πœ‘ β†’ (π»β€˜π‘‹) ∈ 𝐡)
31 fvco3 6991 . . . . . . . 8 ((𝐻:π‘ŒβŸΆπ΅ ∧ 𝑋 ∈ π‘Œ) β†’ ((𝐹 ∘ 𝐻)β€˜π‘‹) = (πΉβ€˜(π»β€˜π‘‹)))
3212, 28, 31syl2anc 582 . . . . . . 7 (πœ‘ β†’ ((𝐹 ∘ 𝐻)β€˜π‘‹) = (πΉβ€˜(π»β€˜π‘‹)))
3313fveq1d 6893 . . . . . . 7 (πœ‘ β†’ ((𝐹 ∘ 𝐻)β€˜π‘‹) = (πΊβ€˜π‘‹))
3432, 33eqtr3d 2767 . . . . . 6 (πœ‘ β†’ (πΉβ€˜(π»β€˜π‘‹)) = (πΊβ€˜π‘‹))
35 cvmlift3lem7.1 . . . . . 6 (πœ‘ β†’ (πΊβ€˜π‘‹) ∈ 𝐴)
3634, 35eqeltrd 2825 . . . . 5 (πœ‘ β†’ (πΉβ€˜(π»β€˜π‘‹)) ∈ 𝐴)
37 cvmlift3lem7.w . . . . . 6 π‘Š = (℩𝑏 ∈ 𝑇 (π»β€˜π‘‹) ∈ 𝑏)
383, 1, 37cvmsiota 34943 . . . . 5 ((𝐹 ∈ (𝐢 CovMap 𝐽) ∧ (𝑇 ∈ (π‘†β€˜π΄) ∧ (π»β€˜π‘‹) ∈ 𝐡 ∧ (πΉβ€˜(π»β€˜π‘‹)) ∈ 𝐴)) β†’ (π‘Š ∈ 𝑇 ∧ (π»β€˜π‘‹) ∈ π‘Š))
394, 29, 30, 36, 38syl13anc 1369 . . . 4 (πœ‘ β†’ (π‘Š ∈ 𝑇 ∧ (π»β€˜π‘‹) ∈ π‘Š))
40 eqid 2725 . . . . . . . . . . 11 (π»β€˜π‘‹) = (π»β€˜π‘‹)
411, 2, 4, 5, 6, 7, 8, 9, 10, 11cvmlift3lem4 34988 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑋 ∈ π‘Œ) β†’ ((π»β€˜π‘‹) = (π»β€˜π‘‹) ↔ βˆƒπ‘“ ∈ (II Cn 𝐾)((π‘“β€˜0) = 𝑂 ∧ (π‘“β€˜1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐢)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (π‘”β€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹))))
4240, 41mpbii 232 . . . . . . . . . 10 ((πœ‘ ∧ 𝑋 ∈ π‘Œ) β†’ βˆƒπ‘“ ∈ (II Cn 𝐾)((π‘“β€˜0) = 𝑂 ∧ (π‘“β€˜1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐢)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (π‘”β€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)))
4328, 42mpdan 685 . . . . . . . . 9 (πœ‘ β†’ βˆƒπ‘“ ∈ (II Cn 𝐾)((π‘“β€˜0) = 𝑂 ∧ (π‘“β€˜1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐢)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (π‘”β€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)))
4443adantr 479 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ 𝑀) β†’ βˆƒπ‘“ ∈ (II Cn 𝐾)((π‘“β€˜0) = 𝑂 ∧ (π‘“β€˜1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐢)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (π‘”β€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)))
45 fveq1 6890 . . . . . . . . . . 11 (𝑓 = β„Ž β†’ (π‘“β€˜0) = (β„Žβ€˜0))
4645eqeq1d 2727 . . . . . . . . . 10 (𝑓 = β„Ž β†’ ((π‘“β€˜0) = 𝑂 ↔ (β„Žβ€˜0) = 𝑂))
47 fveq1 6890 . . . . . . . . . . 11 (𝑓 = β„Ž β†’ (π‘“β€˜1) = (β„Žβ€˜1))
4847eqeq1d 2727 . . . . . . . . . 10 (𝑓 = β„Ž β†’ ((π‘“β€˜1) = 𝑋 ↔ (β„Žβ€˜1) = 𝑋))
49 coeq2 5855 . . . . . . . . . . . . . . . 16 (𝑓 = β„Ž β†’ (𝐺 ∘ 𝑓) = (𝐺 ∘ β„Ž))
5049eqeq2d 2736 . . . . . . . . . . . . . . 15 (𝑓 = β„Ž β†’ ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ β„Ž)))
5150anbi1d 629 . . . . . . . . . . . . . 14 (𝑓 = β„Ž β†’ (((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (π‘”β€˜0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ β„Ž) ∧ (π‘”β€˜0) = 𝑃)))
5251riotabidv 7373 . . . . . . . . . . . . 13 (𝑓 = β„Ž β†’ (℩𝑔 ∈ (II Cn 𝐢)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (π‘”β€˜0) = 𝑃)) = (℩𝑔 ∈ (II Cn 𝐢)((𝐹 ∘ 𝑔) = (𝐺 ∘ β„Ž) ∧ (π‘”β€˜0) = 𝑃)))
53 coeq2 5855 . . . . . . . . . . . . . . . 16 (π‘Ž = 𝑔 β†’ (𝐹 ∘ π‘Ž) = (𝐹 ∘ 𝑔))
5453eqeq1d 2727 . . . . . . . . . . . . . . 15 (π‘Ž = 𝑔 β†’ ((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ β„Ž)))
55 fveq1 6890 . . . . . . . . . . . . . . . 16 (π‘Ž = 𝑔 β†’ (π‘Žβ€˜0) = (π‘”β€˜0))
5655eqeq1d 2727 . . . . . . . . . . . . . . 15 (π‘Ž = 𝑔 β†’ ((π‘Žβ€˜0) = 𝑃 ↔ (π‘”β€˜0) = 𝑃))
5754, 56anbi12d 630 . . . . . . . . . . . . . 14 (π‘Ž = 𝑔 β†’ (((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ β„Ž) ∧ (π‘”β€˜0) = 𝑃)))
5857cbvriotavw 7381 . . . . . . . . . . . . 13 (β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃)) = (℩𝑔 ∈ (II Cn 𝐢)((𝐹 ∘ 𝑔) = (𝐺 ∘ β„Ž) ∧ (π‘”β€˜0) = 𝑃))
5952, 58eqtr4di 2783 . . . . . . . . . . . 12 (𝑓 = β„Ž β†’ (℩𝑔 ∈ (II Cn 𝐢)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (π‘”β€˜0) = 𝑃)) = (β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃)))
6059fveq1d 6893 . . . . . . . . . . 11 (𝑓 = β„Ž β†’ ((℩𝑔 ∈ (II Cn 𝐢)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (π‘”β€˜0) = 𝑃))β€˜1) = ((β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃))β€˜1))
6160eqeq1d 2727 . . . . . . . . . 10 (𝑓 = β„Ž β†’ (((℩𝑔 ∈ (II Cn 𝐢)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (π‘”β€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹) ↔ ((β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)))
6246, 48, 613anbi123d 1432 . . . . . . . . 9 (𝑓 = β„Ž β†’ (((π‘“β€˜0) = 𝑂 ∧ (π‘“β€˜1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐢)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (π‘”β€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)) ↔ ((β„Žβ€˜0) = 𝑂 ∧ (β„Žβ€˜1) = 𝑋 ∧ ((β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹))))
6362cbvrexvw 3226 . . . . . . . 8 (βˆƒπ‘“ ∈ (II Cn 𝐾)((π‘“β€˜0) = 𝑂 ∧ (π‘“β€˜1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐢)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (π‘”β€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)) ↔ βˆƒβ„Ž ∈ (II Cn 𝐾)((β„Žβ€˜0) = 𝑂 ∧ (β„Žβ€˜1) = 𝑋 ∧ ((β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)))
6444, 63sylib 217 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ 𝑀) β†’ βˆƒβ„Ž ∈ (II Cn 𝐾)((β„Žβ€˜0) = 𝑂 ∧ (β„Žβ€˜1) = 𝑋 ∧ ((β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)))
65 cvmlift3lem7.7 . . . . . . . . 9 (πœ‘ β†’ (𝐾 β†Ύt 𝑀) ∈ PConn)
6665adantr 479 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ 𝑀) β†’ (𝐾 β†Ύt 𝑀) ∈ PConn)
672restuni 23082 . . . . . . . . . . 11 ((𝐾 ∈ Top ∧ 𝑀 βŠ† π‘Œ) β†’ 𝑀 = βˆͺ (𝐾 β†Ύt 𝑀))
6816, 24, 67syl2anc 582 . . . . . . . . . 10 (πœ‘ β†’ 𝑀 = βˆͺ (𝐾 β†Ύt 𝑀))
6927, 68eleqtrd 2827 . . . . . . . . 9 (πœ‘ β†’ 𝑋 ∈ βˆͺ (𝐾 β†Ύt 𝑀))
7069adantr 479 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ 𝑀) β†’ 𝑋 ∈ βˆͺ (𝐾 β†Ύt 𝑀))
7168eleq2d 2811 . . . . . . . . 9 (πœ‘ β†’ (𝑦 ∈ 𝑀 ↔ 𝑦 ∈ βˆͺ (𝐾 β†Ύt 𝑀)))
7271biimpa 475 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ 𝑀) β†’ 𝑦 ∈ βˆͺ (𝐾 β†Ύt 𝑀))
73 eqid 2725 . . . . . . . . 9 βˆͺ (𝐾 β†Ύt 𝑀) = βˆͺ (𝐾 β†Ύt 𝑀)
7473pconncn 34890 . . . . . . . 8 (((𝐾 β†Ύt 𝑀) ∈ PConn ∧ 𝑋 ∈ βˆͺ (𝐾 β†Ύt 𝑀) ∧ 𝑦 ∈ βˆͺ (𝐾 β†Ύt 𝑀)) β†’ βˆƒπ‘› ∈ (II Cn (𝐾 β†Ύt 𝑀))((π‘›β€˜0) = 𝑋 ∧ (π‘›β€˜1) = 𝑦))
7566, 70, 72, 74syl3anc 1368 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ 𝑀) β†’ βˆƒπ‘› ∈ (II Cn (𝐾 β†Ύt 𝑀))((π‘›β€˜0) = 𝑋 ∧ (π‘›β€˜1) = 𝑦))
76 reeanv 3217 . . . . . . . 8 (βˆƒβ„Ž ∈ (II Cn 𝐾)βˆƒπ‘› ∈ (II Cn (𝐾 β†Ύt 𝑀))(((β„Žβ€˜0) = 𝑂 ∧ (β„Žβ€˜1) = 𝑋 ∧ ((β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)) ∧ ((π‘›β€˜0) = 𝑋 ∧ (π‘›β€˜1) = 𝑦)) ↔ (βˆƒβ„Ž ∈ (II Cn 𝐾)((β„Žβ€˜0) = 𝑂 ∧ (β„Žβ€˜1) = 𝑋 ∧ ((β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)) ∧ βˆƒπ‘› ∈ (II Cn (𝐾 β†Ύt 𝑀))((π‘›β€˜0) = 𝑋 ∧ (π‘›β€˜1) = 𝑦)))
774ad3antrrr 728 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑦 ∈ 𝑀) ∧ (β„Ž ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 β†Ύt 𝑀)))) ∧ (((β„Žβ€˜0) = 𝑂 ∧ (β„Žβ€˜1) = 𝑋 ∧ ((β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)) ∧ ((π‘›β€˜0) = 𝑋 ∧ (π‘›β€˜1) = 𝑦))) β†’ 𝐹 ∈ (𝐢 CovMap 𝐽))
785ad3antrrr 728 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑦 ∈ 𝑀) ∧ (β„Ž ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 β†Ύt 𝑀)))) ∧ (((β„Žβ€˜0) = 𝑂 ∧ (β„Žβ€˜1) = 𝑋 ∧ ((β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)) ∧ ((π‘›β€˜0) = 𝑋 ∧ (π‘›β€˜1) = 𝑦))) β†’ 𝐾 ∈ SConn)
796ad3antrrr 728 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑦 ∈ 𝑀) ∧ (β„Ž ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 β†Ύt 𝑀)))) ∧ (((β„Žβ€˜0) = 𝑂 ∧ (β„Žβ€˜1) = 𝑋 ∧ ((β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)) ∧ ((π‘›β€˜0) = 𝑋 ∧ (π‘›β€˜1) = 𝑦))) β†’ 𝐾 ∈ 𝑛-Locally PConn)
807ad3antrrr 728 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑦 ∈ 𝑀) ∧ (β„Ž ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 β†Ύt 𝑀)))) ∧ (((β„Žβ€˜0) = 𝑂 ∧ (β„Žβ€˜1) = 𝑋 ∧ ((β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)) ∧ ((π‘›β€˜0) = 𝑋 ∧ (π‘›β€˜1) = 𝑦))) β†’ 𝑂 ∈ π‘Œ)
818ad3antrrr 728 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑦 ∈ 𝑀) ∧ (β„Ž ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 β†Ύt 𝑀)))) ∧ (((β„Žβ€˜0) = 𝑂 ∧ (β„Žβ€˜1) = 𝑋 ∧ ((β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)) ∧ ((π‘›β€˜0) = 𝑋 ∧ (π‘›β€˜1) = 𝑦))) β†’ 𝐺 ∈ (𝐾 Cn 𝐽))
829ad3antrrr 728 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑦 ∈ 𝑀) ∧ (β„Ž ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 β†Ύt 𝑀)))) ∧ (((β„Žβ€˜0) = 𝑂 ∧ (β„Žβ€˜1) = 𝑋 ∧ ((β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)) ∧ ((π‘›β€˜0) = 𝑋 ∧ (π‘›β€˜1) = 𝑦))) β†’ 𝑃 ∈ 𝐡)
8310ad3antrrr 728 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑦 ∈ 𝑀) ∧ (β„Ž ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 β†Ύt 𝑀)))) ∧ (((β„Žβ€˜0) = 𝑂 ∧ (β„Žβ€˜1) = 𝑋 ∧ ((β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)) ∧ ((π‘›β€˜0) = 𝑋 ∧ (π‘›β€˜1) = 𝑦))) β†’ (πΉβ€˜π‘ƒ) = (πΊβ€˜π‘‚))
8435ad3antrrr 728 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑦 ∈ 𝑀) ∧ (β„Ž ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 β†Ύt 𝑀)))) ∧ (((β„Žβ€˜0) = 𝑂 ∧ (β„Žβ€˜1) = 𝑋 ∧ ((β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)) ∧ ((π‘›β€˜0) = 𝑋 ∧ (π‘›β€˜1) = 𝑦))) β†’ (πΊβ€˜π‘‹) ∈ 𝐴)
8529ad3antrrr 728 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑦 ∈ 𝑀) ∧ (β„Ž ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 β†Ύt 𝑀)))) ∧ (((β„Žβ€˜0) = 𝑂 ∧ (β„Žβ€˜1) = 𝑋 ∧ ((β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)) ∧ ((π‘›β€˜0) = 𝑋 ∧ (π‘›β€˜1) = 𝑦))) β†’ 𝑇 ∈ (π‘†β€˜π΄))
8617ad3antrrr 728 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑦 ∈ 𝑀) ∧ (β„Ž ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 β†Ύt 𝑀)))) ∧ (((β„Žβ€˜0) = 𝑂 ∧ (β„Žβ€˜1) = 𝑋 ∧ ((β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)) ∧ ((π‘›β€˜0) = 𝑋 ∧ (π‘›β€˜1) = 𝑦))) β†’ 𝑀 βŠ† (◑𝐺 β€œ 𝐴))
8727ad3antrrr 728 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑦 ∈ 𝑀) ∧ (β„Ž ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 β†Ύt 𝑀)))) ∧ (((β„Žβ€˜0) = 𝑂 ∧ (β„Žβ€˜1) = 𝑋 ∧ ((β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)) ∧ ((π‘›β€˜0) = 𝑋 ∧ (π‘›β€˜1) = 𝑦))) β†’ 𝑋 ∈ 𝑀)
88 simpllr 774 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑦 ∈ 𝑀) ∧ (β„Ž ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 β†Ύt 𝑀)))) ∧ (((β„Žβ€˜0) = 𝑂 ∧ (β„Žβ€˜1) = 𝑋 ∧ ((β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)) ∧ ((π‘›β€˜0) = 𝑋 ∧ (π‘›β€˜1) = 𝑦))) β†’ 𝑦 ∈ 𝑀)
89 simplrl 775 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑦 ∈ 𝑀) ∧ (β„Ž ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 β†Ύt 𝑀)))) ∧ (((β„Žβ€˜0) = 𝑂 ∧ (β„Žβ€˜1) = 𝑋 ∧ ((β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)) ∧ ((π‘›β€˜0) = 𝑋 ∧ (π‘›β€˜1) = 𝑦))) β†’ β„Ž ∈ (II Cn 𝐾))
90 simprl 769 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑦 ∈ 𝑀) ∧ (β„Ž ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 β†Ύt 𝑀)))) ∧ (((β„Žβ€˜0) = 𝑂 ∧ (β„Žβ€˜1) = 𝑋 ∧ ((β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)) ∧ ((π‘›β€˜0) = 𝑋 ∧ (π‘›β€˜1) = 𝑦))) β†’ ((β„Žβ€˜0) = 𝑂 ∧ (β„Žβ€˜1) = 𝑋 ∧ ((β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)))
91 simplrr 776 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑦 ∈ 𝑀) ∧ (β„Ž ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 β†Ύt 𝑀)))) ∧ (((β„Žβ€˜0) = 𝑂 ∧ (β„Žβ€˜1) = 𝑋 ∧ ((β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)) ∧ ((π‘›β€˜0) = 𝑋 ∧ (π‘›β€˜1) = 𝑦))) β†’ 𝑛 ∈ (II Cn (𝐾 β†Ύt 𝑀)))
92 simprr 771 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑦 ∈ 𝑀) ∧ (β„Ž ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 β†Ύt 𝑀)))) ∧ (((β„Žβ€˜0) = 𝑂 ∧ (β„Žβ€˜1) = 𝑋 ∧ ((β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)) ∧ ((π‘›β€˜0) = 𝑋 ∧ (π‘›β€˜1) = 𝑦))) β†’ ((π‘›β€˜0) = 𝑋 ∧ (π‘›β€˜1) = 𝑦))
9353eqeq1d 2727 . . . . . . . . . . . . 13 (π‘Ž = 𝑔 β†’ ((𝐹 ∘ π‘Ž) = (𝐺 ∘ 𝑛) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑛)))
9455eqeq1d 2727 . . . . . . . . . . . . 13 (π‘Ž = 𝑔 β†’ ((π‘Žβ€˜0) = (π»β€˜π‘‹) ↔ (π‘”β€˜0) = (π»β€˜π‘‹)))
9593, 94anbi12d 630 . . . . . . . . . . . 12 (π‘Ž = 𝑔 β†’ (((𝐹 ∘ π‘Ž) = (𝐺 ∘ 𝑛) ∧ (π‘Žβ€˜0) = (π»β€˜π‘‹)) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑛) ∧ (π‘”β€˜0) = (π»β€˜π‘‹))))
9695cbvriotavw 7381 . . . . . . . . . . 11 (β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ 𝑛) ∧ (π‘Žβ€˜0) = (π»β€˜π‘‹))) = (℩𝑔 ∈ (II Cn 𝐢)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑛) ∧ (π‘”β€˜0) = (π»β€˜π‘‹)))
971, 2, 77, 78, 79, 80, 81, 82, 83, 11, 3, 84, 85, 86, 37, 87, 88, 89, 58, 90, 91, 92, 96cvmlift3lem6 34990 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑦 ∈ 𝑀) ∧ (β„Ž ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 β†Ύt 𝑀)))) ∧ (((β„Žβ€˜0) = 𝑂 ∧ (β„Žβ€˜1) = 𝑋 ∧ ((β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)) ∧ ((π‘›β€˜0) = 𝑋 ∧ (π‘›β€˜1) = 𝑦))) β†’ (π»β€˜π‘¦) ∈ π‘Š)
9897ex 411 . . . . . . . . 9 (((πœ‘ ∧ 𝑦 ∈ 𝑀) ∧ (β„Ž ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 β†Ύt 𝑀)))) β†’ ((((β„Žβ€˜0) = 𝑂 ∧ (β„Žβ€˜1) = 𝑋 ∧ ((β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)) ∧ ((π‘›β€˜0) = 𝑋 ∧ (π‘›β€˜1) = 𝑦)) β†’ (π»β€˜π‘¦) ∈ π‘Š))
9998rexlimdvva 3202 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ 𝑀) β†’ (βˆƒβ„Ž ∈ (II Cn 𝐾)βˆƒπ‘› ∈ (II Cn (𝐾 β†Ύt 𝑀))(((β„Žβ€˜0) = 𝑂 ∧ (β„Žβ€˜1) = 𝑋 ∧ ((β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)) ∧ ((π‘›β€˜0) = 𝑋 ∧ (π‘›β€˜1) = 𝑦)) β†’ (π»β€˜π‘¦) ∈ π‘Š))
10076, 99biimtrrid 242 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ 𝑀) β†’ ((βˆƒβ„Ž ∈ (II Cn 𝐾)((β„Žβ€˜0) = 𝑂 ∧ (β„Žβ€˜1) = 𝑋 ∧ ((β„©π‘Ž ∈ (II Cn 𝐢)((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ∧ (π‘Žβ€˜0) = 𝑃))β€˜1) = (π»β€˜π‘‹)) ∧ βˆƒπ‘› ∈ (II Cn (𝐾 β†Ύt 𝑀))((π‘›β€˜0) = 𝑋 ∧ (π‘›β€˜1) = 𝑦)) β†’ (π»β€˜π‘¦) ∈ π‘Š))
10164, 75, 100mp2and 697 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ 𝑀) β†’ (π»β€˜π‘¦) ∈ π‘Š)
102101ralrimiva 3136 . . . . 5 (πœ‘ β†’ βˆ€π‘¦ ∈ 𝑀 (π»β€˜π‘¦) ∈ π‘Š)
10312ffund 6720 . . . . . 6 (πœ‘ β†’ Fun 𝐻)
10412fdmd 6727 . . . . . . 7 (πœ‘ β†’ dom 𝐻 = π‘Œ)
10524, 104sseqtrrd 4014 . . . . . 6 (πœ‘ β†’ 𝑀 βŠ† dom 𝐻)
106 funimass4 6957 . . . . . 6 ((Fun 𝐻 ∧ 𝑀 βŠ† dom 𝐻) β†’ ((𝐻 β€œ 𝑀) βŠ† π‘Š ↔ βˆ€π‘¦ ∈ 𝑀 (π»β€˜π‘¦) ∈ π‘Š))
107103, 105, 106syl2anc 582 . . . . 5 (πœ‘ β†’ ((𝐻 β€œ 𝑀) βŠ† π‘Š ↔ βˆ€π‘¦ ∈ 𝑀 (π»β€˜π‘¦) ∈ π‘Š))
108102, 107mpbird 256 . . . 4 (πœ‘ β†’ (𝐻 β€œ 𝑀) βŠ† π‘Š)
1091, 2, 3, 4, 12, 14, 16, 28, 29, 39, 24, 108cvmlift2lem9a 34969 . . 3 (πœ‘ β†’ (𝐻 β†Ύ 𝑀) ∈ ((𝐾 β†Ύt 𝑀) Cn 𝐢))
11073cncnpi 23198 . . 3 (((𝐻 β†Ύ 𝑀) ∈ ((𝐾 β†Ύt 𝑀) Cn 𝐢) ∧ 𝑋 ∈ βˆͺ (𝐾 β†Ύt 𝑀)) β†’ (𝐻 β†Ύ 𝑀) ∈ (((𝐾 β†Ύt 𝑀) CnP 𝐢)β€˜π‘‹))
111109, 69, 110syl2anc 582 . 2 (πœ‘ β†’ (𝐻 β†Ύ 𝑀) ∈ (((𝐾 β†Ύt 𝑀) CnP 𝐢)β€˜π‘‹))
112 cvmlift3lem7.4 . . . . 5 (πœ‘ β†’ 𝑉 ∈ 𝐾)
1132ssntr 22978 . . . . 5 (((𝐾 ∈ Top ∧ 𝑀 βŠ† π‘Œ) ∧ (𝑉 ∈ 𝐾 ∧ 𝑉 βŠ† 𝑀)) β†’ 𝑉 βŠ† ((intβ€˜πΎ)β€˜π‘€))
11416, 24, 112, 25, 113syl22anc 837 . . . 4 (πœ‘ β†’ 𝑉 βŠ† ((intβ€˜πΎ)β€˜π‘€))
115114, 26sseldd 3973 . . 3 (πœ‘ β†’ 𝑋 ∈ ((intβ€˜πΎ)β€˜π‘€))
1162, 1cnprest 23209 . . 3 (((𝐾 ∈ Top ∧ 𝑀 βŠ† π‘Œ) ∧ (𝑋 ∈ ((intβ€˜πΎ)β€˜π‘€) ∧ 𝐻:π‘ŒβŸΆπ΅)) β†’ (𝐻 ∈ ((𝐾 CnP 𝐢)β€˜π‘‹) ↔ (𝐻 β†Ύ 𝑀) ∈ (((𝐾 β†Ύt 𝑀) CnP 𝐢)β€˜π‘‹)))
11716, 24, 115, 12, 116syl22anc 837 . 2 (πœ‘ β†’ (𝐻 ∈ ((𝐾 CnP 𝐢)β€˜π‘‹) ↔ (𝐻 β†Ύ 𝑀) ∈ (((𝐾 β†Ύt 𝑀) CnP 𝐢)β€˜π‘‹)))
118111, 117mpbird 256 1 (πœ‘ β†’ 𝐻 ∈ ((𝐾 CnP 𝐢)β€˜π‘‹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3051  βˆƒwrex 3060  {crab 3419   βˆ– cdif 3937   ∩ cin 3939   βŠ† wss 3940  βˆ…c0 4318  π’« cpw 4598  {csn 4624  βˆͺ cuni 4903   ↦ cmpt 5226  β—‘ccnv 5671  dom cdm 5672   β†Ύ cres 5674   β€œ cima 5675   ∘ ccom 5676  Fun wfun 6536  βŸΆwf 6538  β€˜cfv 6542  β„©crio 7370  (class class class)co 7415  0cc0 11136  1c1 11137   β†Ύt crest 17399  Topctop 22811  intcnt 22937   Cn ccn 23144   CnP ccnp 23145  π‘›-Locally cnlly 23385  Homeochmeo 23673  IIcii 24811  PConncpconn 34885  SConncsconn 34886   CovMap ccvm 34921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-inf2 9662  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213  ax-pre-sup 11214  ax-addf 11215
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-of 7681  df-om 7868  df-1st 7989  df-2nd 7990  df-supp 8162  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-1o 8483  df-2o 8484  df-er 8721  df-ec 8723  df-map 8843  df-ixp 8913  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-fsupp 9384  df-fi 9432  df-sup 9463  df-inf 9464  df-oi 9531  df-card 9960  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-div 11900  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12501  df-z 12587  df-dec 12706  df-uz 12851  df-q 12961  df-rp 13005  df-xneg 13122  df-xadd 13123  df-xmul 13124  df-ioo 13358  df-ico 13360  df-icc 13361  df-fz 13515  df-fzo 13658  df-fl 13787  df-seq 13997  df-exp 14057  df-hash 14320  df-cj 15076  df-re 15077  df-im 15078  df-sqrt 15212  df-abs 15213  df-clim 15462  df-sum 15663  df-struct 17113  df-sets 17130  df-slot 17148  df-ndx 17160  df-base 17178  df-ress 17207  df-plusg 17243  df-mulr 17244  df-starv 17245  df-sca 17246  df-vsca 17247  df-ip 17248  df-tset 17249  df-ple 17250  df-ds 17252  df-unif 17253  df-hom 17254  df-cco 17255  df-rest 17401  df-topn 17402  df-0g 17420  df-gsum 17421  df-topgen 17422  df-pt 17423  df-prds 17426  df-xrs 17481  df-qtop 17486  df-imas 17487  df-xps 17489  df-mre 17563  df-mrc 17564  df-acs 17566  df-mgm 18597  df-sgrp 18676  df-mnd 18692  df-submnd 18738  df-mulg 19026  df-cntz 19270  df-cmn 19739  df-psmet 21273  df-xmet 21274  df-met 21275  df-bl 21276  df-mopn 21277  df-cnfld 21282  df-top 22812  df-topon 22829  df-topsp 22851  df-bases 22865  df-cld 22939  df-ntr 22940  df-cls 22941  df-nei 23018  df-cn 23147  df-cnp 23148  df-cmp 23307  df-conn 23332  df-lly 23386  df-nlly 23387  df-tx 23482  df-hmeo 23675  df-xms 24242  df-ms 24243  df-tms 24244  df-ii 24813  df-cncf 24814  df-htpy 24912  df-phtpy 24913  df-phtpc 24934  df-pco 24948  df-pconn 34887  df-sconn 34888  df-cvm 34922
This theorem is referenced by:  cvmlift3lem8  34992
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