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Theorem cvmlift3lem7 33983
Description: Lemma for cvmlift3 33986. (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 33979 . . . 4 (πœ‘ β†’ 𝐻:π‘ŒβŸΆπ΅)
131, 2, 4, 5, 6, 7, 8, 9, 10, 11cvmlift3lem5 33981 . . . . 5 (πœ‘ β†’ (𝐹 ∘ 𝐻) = 𝐺)
1413, 8eqeltrd 2834 . . . 4 (πœ‘ β†’ (𝐹 ∘ 𝐻) ∈ (𝐾 Cn 𝐽))
15 sconntop 33886 . . . . 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 6037 . . . . . . 7 (◑𝐺 β€œ 𝐴) βŠ† dom 𝐺
19 eqid 2733 . . . . . . . . 9 βˆͺ 𝐽 = βˆͺ 𝐽
202, 19cnf 22620 . . . . . . . 8 (𝐺 ∈ (𝐾 Cn 𝐽) β†’ 𝐺:π‘ŒβŸΆβˆͺ 𝐽)
21 fdm 6681 . . . . . . . 8 (𝐺:π‘ŒβŸΆβˆͺ 𝐽 β†’ dom 𝐺 = π‘Œ)
228, 20, 213syl 18 . . . . . . 7 (πœ‘ β†’ dom 𝐺 = π‘Œ)
2318, 22sseqtrid 4000 . . . . . 6 (πœ‘ β†’ (◑𝐺 β€œ 𝐴) βŠ† π‘Œ)
2417, 23sstrd 3958 . . . . 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 3949 . . . . 5 (πœ‘ β†’ 𝑋 ∈ 𝑀)
2824, 27sseldd 3949 . . . 4 (πœ‘ β†’ 𝑋 ∈ π‘Œ)
29 cvmlift3lem7.2 . . . 4 (πœ‘ β†’ 𝑇 ∈ (π‘†β€˜π΄))
3012, 28ffvelcdmd 7040 . . . . 5 (πœ‘ β†’ (π»β€˜π‘‹) ∈ 𝐡)
31 fvco3 6944 . . . . . . . 8 ((𝐻:π‘ŒβŸΆπ΅ ∧ 𝑋 ∈ π‘Œ) β†’ ((𝐹 ∘ 𝐻)β€˜π‘‹) = (πΉβ€˜(π»β€˜π‘‹)))
3212, 28, 31syl2anc 585 . . . . . . 7 (πœ‘ β†’ ((𝐹 ∘ 𝐻)β€˜π‘‹) = (πΉβ€˜(π»β€˜π‘‹)))
3313fveq1d 6848 . . . . . . 7 (πœ‘ β†’ ((𝐹 ∘ 𝐻)β€˜π‘‹) = (πΊβ€˜π‘‹))
3432, 33eqtr3d 2775 . . . . . 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 2834 . . . . 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 33935 . . . . 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 1373 . . . 4 (πœ‘ β†’ (π‘Š ∈ 𝑇 ∧ (π»β€˜π‘‹) ∈ π‘Š))
40 eqid 2733 . . . . . . . . . . 11 (π»β€˜π‘‹) = (π»β€˜π‘‹)
411, 2, 4, 5, 6, 7, 8, 9, 10, 11cvmlift3lem4 33980 . . . . . . . . . . 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 686 . . . . . . . . 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 482 . . . . . . . 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 6845 . . . . . . . . . . 11 (𝑓 = β„Ž β†’ (π‘“β€˜0) = (β„Žβ€˜0))
4645eqeq1d 2735 . . . . . . . . . 10 (𝑓 = β„Ž β†’ ((π‘“β€˜0) = 𝑂 ↔ (β„Žβ€˜0) = 𝑂))
47 fveq1 6845 . . . . . . . . . . 11 (𝑓 = β„Ž β†’ (π‘“β€˜1) = (β„Žβ€˜1))
4847eqeq1d 2735 . . . . . . . . . 10 (𝑓 = β„Ž β†’ ((π‘“β€˜1) = 𝑋 ↔ (β„Žβ€˜1) = 𝑋))
49 coeq2 5818 . . . . . . . . . . . . . . . 16 (𝑓 = β„Ž β†’ (𝐺 ∘ 𝑓) = (𝐺 ∘ β„Ž))
5049eqeq2d 2744 . . . . . . . . . . . . . . 15 (𝑓 = β„Ž β†’ ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ β„Ž)))
5150anbi1d 631 . . . . . . . . . . . . . 14 (𝑓 = β„Ž β†’ (((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (π‘”β€˜0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ β„Ž) ∧ (π‘”β€˜0) = 𝑃)))
5251riotabidv 7319 . . . . . . . . . . . . 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 5818 . . . . . . . . . . . . . . . 16 (π‘Ž = 𝑔 β†’ (𝐹 ∘ π‘Ž) = (𝐹 ∘ 𝑔))
5453eqeq1d 2735 . . . . . . . . . . . . . . 15 (π‘Ž = 𝑔 β†’ ((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ β„Ž)))
55 fveq1 6845 . . . . . . . . . . . . . . . 16 (π‘Ž = 𝑔 β†’ (π‘Žβ€˜0) = (π‘”β€˜0))
5655eqeq1d 2735 . . . . . . . . . . . . . . 15 (π‘Ž = 𝑔 β†’ ((π‘Žβ€˜0) = 𝑃 ↔ (π‘”β€˜0) = 𝑃))
5754, 56anbi12d 632 . . . . . . . . . . . . . 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 7327 . . . . . . . . . . . . 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 2791 . . . . . . . . . . . 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 6848 . . . . . . . . . . 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 2735 . . . . . . . . . 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 1437 . . . . . . . . 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 3225 . . . . . . . 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 482 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ 𝑀) β†’ (𝐾 β†Ύt 𝑀) ∈ PConn)
672restuni 22536 . . . . . . . . . . 11 ((𝐾 ∈ Top ∧ 𝑀 βŠ† π‘Œ) β†’ 𝑀 = βˆͺ (𝐾 β†Ύt 𝑀))
6816, 24, 67syl2anc 585 . . . . . . . . . 10 (πœ‘ β†’ 𝑀 = βˆͺ (𝐾 β†Ύt 𝑀))
6927, 68eleqtrd 2836 . . . . . . . . 9 (πœ‘ β†’ 𝑋 ∈ βˆͺ (𝐾 β†Ύt 𝑀))
7069adantr 482 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ 𝑀) β†’ 𝑋 ∈ βˆͺ (𝐾 β†Ύt 𝑀))
7168eleq2d 2820 . . . . . . . . 9 (πœ‘ β†’ (𝑦 ∈ 𝑀 ↔ 𝑦 ∈ βˆͺ (𝐾 β†Ύt 𝑀)))
7271biimpa 478 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ 𝑀) β†’ 𝑦 ∈ βˆͺ (𝐾 β†Ύt 𝑀))
73 eqid 2733 . . . . . . . . 9 βˆͺ (𝐾 β†Ύt 𝑀) = βˆͺ (𝐾 β†Ύt 𝑀)
7473pconncn 33882 . . . . . . . 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 1372 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ 𝑀) β†’ βˆƒπ‘› ∈ (II Cn (𝐾 β†Ύt 𝑀))((π‘›β€˜0) = 𝑋 ∧ (π‘›β€˜1) = 𝑦))
76 reeanv 3216 . . . . . . . 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 729 . . . . . . . . . . 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 729 . . . . . . . . . . 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 729 . . . . . . . . . . 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 729 . . . . . . . . . . 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 729 . . . . . . . . . . 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 729 . . . . . . . . . . 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 729 . . . . . . . . . . 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 729 . . . . . . . . . . 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 729 . . . . . . . . . . 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 729 . . . . . . . . . . 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 729 . . . . . . . . . . 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 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†’ 𝑦 ∈ 𝑀)
89 simplrl 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†’ β„Ž ∈ (II Cn 𝐾))
90 simprl 770 . . . . . . . . . . 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 777 . . . . . . . . . . 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 772 . . . . . . . . . . 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 2735 . . . . . . . . . . . . 13 (π‘Ž = 𝑔 β†’ ((𝐹 ∘ π‘Ž) = (𝐺 ∘ 𝑛) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑛)))
9455eqeq1d 2735 . . . . . . . . . . . . 13 (π‘Ž = 𝑔 β†’ ((π‘Žβ€˜0) = (π»β€˜π‘‹) ↔ (π‘”β€˜0) = (π»β€˜π‘‹)))
9593, 94anbi12d 632 . . . . . . . . . . . 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 7327 . . . . . . . . . . 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 33982 . . . . . . . . . 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 414 . . . . . . . . 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 698 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ 𝑀) β†’ (π»β€˜π‘¦) ∈ π‘Š)
102101ralrimiva 3140 . . . . 5 (πœ‘ β†’ βˆ€π‘¦ ∈ 𝑀 (π»β€˜π‘¦) ∈ π‘Š)
10312ffund 6676 . . . . . 6 (πœ‘ β†’ Fun 𝐻)
10412fdmd 6683 . . . . . . 7 (πœ‘ β†’ dom 𝐻 = π‘Œ)
10524, 104sseqtrrd 3989 . . . . . 6 (πœ‘ β†’ 𝑀 βŠ† dom 𝐻)
106 funimass4 6911 . . . . . 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 585 . . . . 5 (πœ‘ β†’ ((𝐻 β€œ 𝑀) βŠ† π‘Š ↔ βˆ€π‘¦ ∈ 𝑀 (π»β€˜π‘¦) ∈ π‘Š))
108102, 107mpbird 257 . . . 4 (πœ‘ β†’ (𝐻 β€œ 𝑀) βŠ† π‘Š)
1091, 2, 3, 4, 12, 14, 16, 28, 29, 39, 24, 108cvmlift2lem9a 33961 . . 3 (πœ‘ β†’ (𝐻 β†Ύ 𝑀) ∈ ((𝐾 β†Ύt 𝑀) Cn 𝐢))
11073cncnpi 22652 . . 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 585 . 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 22432 . . . . 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 838 . . . 4 (πœ‘ β†’ 𝑉 βŠ† ((intβ€˜πΎ)β€˜π‘€))
115114, 26sseldd 3949 . . 3 (πœ‘ β†’ 𝑋 ∈ ((intβ€˜πΎ)β€˜π‘€))
1162, 1cnprest 22663 . . 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 838 . 2 (πœ‘ β†’ (𝐻 ∈ ((𝐾 CnP 𝐢)β€˜π‘‹) ↔ (𝐻 β†Ύ 𝑀) ∈ (((𝐾 β†Ύt 𝑀) CnP 𝐢)β€˜π‘‹)))
118111, 117mpbird 257 1 (πœ‘ β†’ 𝐻 ∈ ((𝐾 CnP 𝐢)β€˜π‘‹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070  {crab 3406   βˆ– cdif 3911   ∩ cin 3913   βŠ† wss 3914  βˆ…c0 4286  π’« cpw 4564  {csn 4590  βˆͺ cuni 4869   ↦ cmpt 5192  β—‘ccnv 5636  dom cdm 5637   β†Ύ cres 5639   β€œ cima 5640   ∘ ccom 5641  Fun wfun 6494  βŸΆwf 6496  β€˜cfv 6500  β„©crio 7316  (class class class)co 7361  0cc0 11059  1c1 11060   β†Ύt crest 17310  Topctop 22265  intcnt 22391   Cn ccn 22598   CnP ccnp 22599  π‘›-Locally cnlly 22839  Homeochmeo 23127  IIcii 24261  PConncpconn 33877  SConncsconn 33878   CovMap ccvm 33913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-inf2 9585  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137  ax-addf 11138  ax-mulf 11139
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-se 5593  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-of 7621  df-om 7807  df-1st 7925  df-2nd 7926  df-supp 8097  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-2o 8417  df-er 8654  df-ec 8656  df-map 8773  df-ixp 8842  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-fsupp 9312  df-fi 9355  df-sup 9386  df-inf 9387  df-oi 9454  df-card 9883  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-div 11821  df-nn 12162  df-2 12224  df-3 12225  df-4 12226  df-5 12227  df-6 12228  df-7 12229  df-8 12230  df-9 12231  df-n0 12422  df-z 12508  df-dec 12627  df-uz 12772  df-q 12882  df-rp 12924  df-xneg 13041  df-xadd 13042  df-xmul 13043  df-ioo 13277  df-ico 13279  df-icc 13280  df-fz 13434  df-fzo 13577  df-fl 13706  df-seq 13916  df-exp 13977  df-hash 14240  df-cj 14993  df-re 14994  df-im 14995  df-sqrt 15129  df-abs 15130  df-clim 15379  df-sum 15580  df-struct 17027  df-sets 17044  df-slot 17062  df-ndx 17074  df-base 17092  df-ress 17121  df-plusg 17154  df-mulr 17155  df-starv 17156  df-sca 17157  df-vsca 17158  df-ip 17159  df-tset 17160  df-ple 17161  df-ds 17163  df-unif 17164  df-hom 17165  df-cco 17166  df-rest 17312  df-topn 17313  df-0g 17331  df-gsum 17332  df-topgen 17333  df-pt 17334  df-prds 17337  df-xrs 17392  df-qtop 17397  df-imas 17398  df-xps 17400  df-mre 17474  df-mrc 17475  df-acs 17477  df-mgm 18505  df-sgrp 18554  df-mnd 18565  df-submnd 18610  df-mulg 18881  df-cntz 19105  df-cmn 19572  df-psmet 20811  df-xmet 20812  df-met 20813  df-bl 20814  df-mopn 20815  df-cnfld 20820  df-top 22266  df-topon 22283  df-topsp 22305  df-bases 22319  df-cld 22393  df-ntr 22394  df-cls 22395  df-nei 22472  df-cn 22601  df-cnp 22602  df-cmp 22761  df-conn 22786  df-lly 22840  df-nlly 22841  df-tx 22936  df-hmeo 23129  df-xms 23696  df-ms 23697  df-tms 23698  df-ii 24263  df-htpy 24356  df-phtpy 24357  df-phtpc 24378  df-pco 24391  df-pconn 33879  df-sconn 33880  df-cvm 33914
This theorem is referenced by:  cvmlift3lem8  33984
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