cvmsiota - Mathbox for Mario Carneiro

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Theorem cvmsiota 34796

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 34795	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → ∃!𝑥 ∈ 𝑇 𝐴 ∈ 𝑥)
5		riotacl2 7378	. . . 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 2831	. 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → 𝑊 ∈ {𝑥 ∈ 𝑇 ∣ 𝐴 ∈ 𝑥})
8		eleq2 2816	. . 3 ⊢ (𝑣 = 𝑊 → (𝐴 ∈ 𝑣 ↔ 𝐴 ∈ 𝑊))
9		eleq2 2816	. . . 4 ⊢ (𝑥 = 𝑣 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑣))
10	9	cbvrabv 3436	. . 3 ⊢ {𝑥 ∈ 𝑇 ∣ 𝐴 ∈ 𝑥} = {𝑣 ∈ 𝑇 ∣ 𝐴 ∈ 𝑣}
11	8, 10	elrab2 3681	. 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 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ∃!wreu 3368 {crab 3426 ∖ cdif 3940 ∩ cin 3942 ∅c0 4317 𝒫 cpw 4597 {csn 4623 ∪ cuni 4902 ↦ cmpt 5224 ^◡ccnv 5668 ↾ cres 5671 “ cima 5672 ‘cfv 6537 ℩crio 7360 (class class class)co 7405 ↾_t crest 17375 Homeochmeo 23612 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-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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-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-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 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-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-map 8824 df-top 22751 df-topon 22768 df-cn 23086 df-cvm 34775

This theorem is referenced by: cvmopnlem 34797 cvmliftmolem2 34801 cvmliftlem6 34809 cvmliftlem8 34811 cvmliftlem9 34812 cvmlift2lem9 34830 cvmlift3lem6 34843 cvmlift3lem7 34844

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