cvmsiota - Mathbox for Mario Carneiro

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Theorem cvmsiota 33139

Description: Identify the unique element of 𝑇 containing 𝐴. (Contributed by Mario Carneiro, 14-Feb-2015.)

**Hypotheses**
Ref	Expression
cvmcov.1	⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (^◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾_t 𝑢)Homeo(𝐽 ↾_t 𝑘))))})
cvmseu.1	⊢ 𝐵 = ∪ 𝐶
cvmsiota.2	⊢ 𝑊 = (℩𝑥 ∈ 𝑇 𝐴 ∈ 𝑥)

**Assertion**
Ref	Expression
cvmsiota	⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → (𝑊 ∈ 𝑇 ∧ 𝐴 ∈ 𝑊))

Distinct variable groups: 𝑘,𝑠,𝑢,𝑣,𝑥,𝐶 𝑘,𝐹,𝑠,𝑢,𝑣,𝑥 𝑘,𝐽,𝑠,𝑢,𝑣,𝑥 𝑥,𝑆 𝑈,𝑘,𝑠,𝑢,𝑣,𝑥 𝑇,𝑠,𝑢,𝑣,𝑥 𝑣,𝑊 𝑢,𝐴,𝑣,𝑥 𝑣,𝐵,𝑥 Allowed substitution hints: 𝐴(𝑘,𝑠) 𝐵(𝑢,𝑘,𝑠) 𝑆(𝑣,𝑢,𝑘,𝑠) 𝑇(𝑘) 𝑊(𝑥,𝑢,𝑘,𝑠)

**Proof of Theorem cvmsiota**
Step	Hyp	Ref	Expression
1		cvmsiota.2	. . 3 ⊢ 𝑊 = (℩𝑥 ∈ 𝑇 𝐴 ∈ 𝑥)
2		cvmcov.1	. . . . 5 ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (^◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾_t 𝑢)Homeo(𝐽 ↾_t 𝑘))))})
3		cvmseu.1	. . . . 5 ⊢ 𝐵 = ∪ 𝐶
4	2, 3	cvmseu 33138	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → ∃!𝑥 ∈ 𝑇 𝐴 ∈ 𝑥)
5		riotacl2 7229	. . . 4 ⊢ (∃!𝑥 ∈ 𝑇 𝐴 ∈ 𝑥 → (℩𝑥 ∈ 𝑇 𝐴 ∈ 𝑥) ∈ {𝑥 ∈ 𝑇 ∣ 𝐴 ∈ 𝑥})
6	4, 5	syl 17	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → (℩𝑥 ∈ 𝑇 𝐴 ∈ 𝑥) ∈ {𝑥 ∈ 𝑇 ∣ 𝐴 ∈ 𝑥})
7	1, 6	eqeltrid 2843	. 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → 𝑊 ∈ {𝑥 ∈ 𝑇 ∣ 𝐴 ∈ 𝑥})
8		eleq2 2827	. . 3 ⊢ (𝑣 = 𝑊 → (𝐴 ∈ 𝑣 ↔ 𝐴 ∈ 𝑊))
9		eleq2 2827	. . . 4 ⊢ (𝑥 = 𝑣 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑣))
10	9	cbvrabv 3416	. . 3 ⊢ {𝑥 ∈ 𝑇 ∣ 𝐴 ∈ 𝑥} = {𝑣 ∈ 𝑇 ∣ 𝐴 ∈ 𝑣}
11	8, 10	elrab2 3620	. 2 ⊢ (𝑊 ∈ {𝑥 ∈ 𝑇 ∣ 𝐴 ∈ 𝑥} ↔ (𝑊 ∈ 𝑇 ∧ 𝐴 ∈ 𝑊))
12	7, 11	sylib 217	1 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → (𝑊 ∈ 𝑇 ∧ 𝐴 ∈ 𝑊))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ∀wral 3063 ∃!wreu 3065 {crab 3067 ∖ cdif 3880 ∩ cin 3882 ∅c0 4253 𝒫 cpw 4530 {csn 4558 ∪ cuni 4836 ↦ cmpt 5153 ^◡ccnv 5579 ↾ cres 5582 “ cima 5583 ‘cfv 6418 ℩crio 7211 (class class class)co 7255 ↾_t crest 17048 Homeochmeo 22812 CovMap ccvm 33117

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-map 8575 df-top 21951 df-topon 21968 df-cn 22286 df-cvm 33118

This theorem is referenced by: cvmopnlem 33140 cvmliftmolem2 33144 cvmliftlem6 33152 cvmliftlem8 33154 cvmliftlem9 33155 cvmlift2lem9 33173 cvmlift3lem6 33186 cvmlift3lem7 33187

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