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Theorem cvmlift3lem7 34844
Description: Lemma for cvmlift3 34847. (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 34840 . . . 4 (πœ‘ β†’ 𝐻:π‘ŒβŸΆπ΅)
131, 2, 4, 5, 6, 7, 8, 9, 10, 11cvmlift3lem5 34842 . . . . 5 (πœ‘ β†’ (𝐹 ∘ 𝐻) = 𝐺)
1413, 8eqeltrd 2827 . . . 4 (πœ‘ β†’ (𝐹 ∘ 𝐻) ∈ (𝐾 Cn 𝐽))
15 sconntop 34747 . . . . 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 6074 . . . . . . 7 (◑𝐺 β€œ 𝐴) βŠ† dom 𝐺
19 eqid 2726 . . . . . . . . 9 βˆͺ 𝐽 = βˆͺ 𝐽
202, 19cnf 23105 . . . . . . . 8 (𝐺 ∈ (𝐾 Cn 𝐽) β†’ 𝐺:π‘ŒβŸΆβˆͺ 𝐽)
21 fdm 6720 . . . . . . . 8 (𝐺:π‘ŒβŸΆβˆͺ 𝐽 β†’ dom 𝐺 = π‘Œ)
228, 20, 213syl 18 . . . . . . 7 (πœ‘ β†’ dom 𝐺 = π‘Œ)
2318, 22sseqtrid 4029 . . . . . 6 (πœ‘ β†’ (◑𝐺 β€œ 𝐴) βŠ† π‘Œ)
2417, 23sstrd 3987 . . . . 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 3978 . . . . 5 (πœ‘ β†’ 𝑋 ∈ 𝑀)
2824, 27sseldd 3978 . . . 4 (πœ‘ β†’ 𝑋 ∈ π‘Œ)
29 cvmlift3lem7.2 . . . 4 (πœ‘ β†’ 𝑇 ∈ (π‘†β€˜π΄))
3012, 28ffvelcdmd 7081 . . . . 5 (πœ‘ β†’ (π»β€˜π‘‹) ∈ 𝐡)
31 fvco3 6984 . . . . . . . 8 ((𝐻:π‘ŒβŸΆπ΅ ∧ 𝑋 ∈ π‘Œ) β†’ ((𝐹 ∘ 𝐻)β€˜π‘‹) = (πΉβ€˜(π»β€˜π‘‹)))
3212, 28, 31syl2anc 583 . . . . . . 7 (πœ‘ β†’ ((𝐹 ∘ 𝐻)β€˜π‘‹) = (πΉβ€˜(π»β€˜π‘‹)))
3313fveq1d 6887 . . . . . . 7 (πœ‘ β†’ ((𝐹 ∘ 𝐻)β€˜π‘‹) = (πΊβ€˜π‘‹))
3432, 33eqtr3d 2768 . . . . . 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 2827 . . . . 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 34796 . . . . 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 2726 . . . . . . . . . . 11 (π»β€˜π‘‹) = (π»β€˜π‘‹)
411, 2, 4, 5, 6, 7, 8, 9, 10, 11cvmlift3lem4 34841 . . . . . . . . . . 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 684 . . . . . . . . 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 480 . . . . . . . 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 6884 . . . . . . . . . . 11 (𝑓 = β„Ž β†’ (π‘“β€˜0) = (β„Žβ€˜0))
4645eqeq1d 2728 . . . . . . . . . 10 (𝑓 = β„Ž β†’ ((π‘“β€˜0) = 𝑂 ↔ (β„Žβ€˜0) = 𝑂))
47 fveq1 6884 . . . . . . . . . . 11 (𝑓 = β„Ž β†’ (π‘“β€˜1) = (β„Žβ€˜1))
4847eqeq1d 2728 . . . . . . . . . 10 (𝑓 = β„Ž β†’ ((π‘“β€˜1) = 𝑋 ↔ (β„Žβ€˜1) = 𝑋))
49 coeq2 5852 . . . . . . . . . . . . . . . 16 (𝑓 = β„Ž β†’ (𝐺 ∘ 𝑓) = (𝐺 ∘ β„Ž))
5049eqeq2d 2737 . . . . . . . . . . . . . . 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 7363 . . . . . . . . . . . . 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 5852 . . . . . . . . . . . . . . . 16 (π‘Ž = 𝑔 β†’ (𝐹 ∘ π‘Ž) = (𝐹 ∘ 𝑔))
5453eqeq1d 2728 . . . . . . . . . . . . . . 15 (π‘Ž = 𝑔 β†’ ((𝐹 ∘ π‘Ž) = (𝐺 ∘ β„Ž) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ β„Ž)))
55 fveq1 6884 . . . . . . . . . . . . . . . 16 (π‘Ž = 𝑔 β†’ (π‘Žβ€˜0) = (π‘”β€˜0))
5655eqeq1d 2728 . . . . . . . . . . . . . . 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 7371 . . . . . . . . . . . . 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 2784 . . . . . . . . . . . 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 6887 . . . . . . . . . . 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 2728 . . . . . . . . . 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 3229 . . . . . . . 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 480 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ 𝑀) β†’ (𝐾 β†Ύt 𝑀) ∈ PConn)
672restuni 23021 . . . . . . . . . . 11 ((𝐾 ∈ Top ∧ 𝑀 βŠ† π‘Œ) β†’ 𝑀 = βˆͺ (𝐾 β†Ύt 𝑀))
6816, 24, 67syl2anc 583 . . . . . . . . . 10 (πœ‘ β†’ 𝑀 = βˆͺ (𝐾 β†Ύt 𝑀))
6927, 68eleqtrd 2829 . . . . . . . . 9 (πœ‘ β†’ 𝑋 ∈ βˆͺ (𝐾 β†Ύt 𝑀))
7069adantr 480 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ 𝑀) β†’ 𝑋 ∈ βˆͺ (𝐾 β†Ύt 𝑀))
7168eleq2d 2813 . . . . . . . . 9 (πœ‘ β†’ (𝑦 ∈ 𝑀 ↔ 𝑦 ∈ βˆͺ (𝐾 β†Ύt 𝑀)))
7271biimpa 476 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ 𝑀) β†’ 𝑦 ∈ βˆͺ (𝐾 β†Ύt 𝑀))
73 eqid 2726 . . . . . . . . 9 βˆͺ (𝐾 β†Ύt 𝑀) = βˆͺ (𝐾 β†Ύt 𝑀)
7473pconncn 34743 . . . . . . . 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 3220 . . . . . . . 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 727 . . . . . . . . . . 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 727 . . . . . . . . . . 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 727 . . . . . . . . . . 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 727 . . . . . . . . . . 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 727 . . . . . . . . . . 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 727 . . . . . . . . . . 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 727 . . . . . . . . . . 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 727 . . . . . . . . . . 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 727 . . . . . . . . . . 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 727 . . . . . . . . . . 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 727 . . . . . . . . . . 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 773 . . . . . . . . . . 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 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†’ β„Ž ∈ (II Cn 𝐾))
90 simprl 768 . . . . . . . . . . 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 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†’ 𝑛 ∈ (II Cn (𝐾 β†Ύt 𝑀)))
92 simprr 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†’ ((π‘›β€˜0) = 𝑋 ∧ (π‘›β€˜1) = 𝑦))
9353eqeq1d 2728 . . . . . . . . . . . . 13 (π‘Ž = 𝑔 β†’ ((𝐹 ∘ π‘Ž) = (𝐺 ∘ 𝑛) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑛)))
9455eqeq1d 2728 . . . . . . . . . . . . 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 7371 . . . . . . . . . . 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 34843 . . . . . . . . . 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 412 . . . . . . . . 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 3205 . . . . . . . 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 696 . . . . . 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 6715 . . . . . 6 (πœ‘ β†’ Fun 𝐻)
10412fdmd 6722 . . . . . . 7 (πœ‘ β†’ dom 𝐻 = π‘Œ)
10524, 104sseqtrrd 4018 . . . . . 6 (πœ‘ β†’ 𝑀 βŠ† dom 𝐻)
106 funimass4 6950 . . . . . 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 583 . . . . 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 34822 . . 3 (πœ‘ β†’ (𝐻 β†Ύ 𝑀) ∈ ((𝐾 β†Ύt 𝑀) Cn 𝐢))
11073cncnpi 23137 . . 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 583 . 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 22917 . . . . 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 836 . . . 4 (πœ‘ β†’ 𝑉 βŠ† ((intβ€˜πΎ)β€˜π‘€))
115114, 26sseldd 3978 . . 3 (πœ‘ β†’ 𝑋 ∈ ((intβ€˜πΎ)β€˜π‘€))
1162, 1cnprest 23148 . . 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 836 . 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  βˆƒwrex 3064  {crab 3426   βˆ– cdif 3940   ∩ cin 3942   βŠ† wss 3943  βˆ…c0 4317  π’« cpw 4597  {csn 4623  βˆͺ cuni 4902   ↦ cmpt 5224  β—‘ccnv 5668  dom cdm 5669   β†Ύ cres 5671   β€œ cima 5672   ∘ ccom 5673  Fun wfun 6531  βŸΆwf 6533  β€˜cfv 6537  β„©crio 7360  (class class class)co 7405  0cc0 11112  1c1 11113   β†Ύt crest 17375  Topctop 22750  intcnt 22876   Cn ccn 23083   CnP ccnp 23084  π‘›-Locally cnlly 23324  Homeochmeo 23612  IIcii 24750  PConncpconn 34738  SConncsconn 34739   CovMap ccvm 34774
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190  ax-addf 11191
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7667  df-om 7853  df-1st 7974  df-2nd 7975  df-supp 8147  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-2o 8468  df-er 8705  df-ec 8707  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-fi 9408  df-sup 9439  df-inf 9440  df-oi 9507  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-q 12937  df-rp 12981  df-xneg 13098  df-xadd 13099  df-xmul 13100  df-ioo 13334  df-ico 13336  df-icc 13337  df-fz 13491  df-fzo 13634  df-fl 13763  df-seq 13973  df-exp 14033  df-hash 14296  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-clim 15438  df-sum 15639  df-struct 17089  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-mulr 17220  df-starv 17221  df-sca 17222  df-vsca 17223  df-ip 17224  df-tset 17225  df-ple 17226  df-ds 17228  df-unif 17229  df-hom 17230  df-cco 17231  df-rest 17377  df-topn 17378  df-0g 17396  df-gsum 17397  df-topgen 17398  df-pt 17399  df-prds 17402  df-xrs 17457  df-qtop 17462  df-imas 17463  df-xps 17465  df-mre 17539  df-mrc 17540  df-acs 17542  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-submnd 18714  df-mulg 18996  df-cntz 19233  df-cmn 19702  df-psmet 21232  df-xmet 21233  df-met 21234  df-bl 21235  df-mopn 21236  df-cnfld 21241  df-top 22751  df-topon 22768  df-topsp 22790  df-bases 22804  df-cld 22878  df-ntr 22879  df-cls 22880  df-nei 22957  df-cn 23086  df-cnp 23087  df-cmp 23246  df-conn 23271  df-lly 23325  df-nlly 23326  df-tx 23421  df-hmeo 23614  df-xms 24181  df-ms 24182  df-tms 24183  df-ii 24752  df-cncf 24753  df-htpy 24851  df-phtpy 24852  df-phtpc 24873  df-pco 24887  df-pconn 34740  df-sconn 34741  df-cvm 34775
This theorem is referenced by:  cvmlift3lem8  34845
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