cvmsiota - Mathbox for Mario Carneiro

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Theorem cvmsiota 35856

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 35855	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → ∃!𝑥 ∈ 𝑇 𝐴 ∈ 𝑥)
5		riotacl2 7386	. . . 4 ⊢ (∃!𝑥 ∈ 𝑇 𝐴 ∈ 𝑥 → (℩𝑥 ∈ 𝑇 𝐴 ∈ 𝑥) ∈ {𝑥 ∈ 𝑇 ∣ 𝐴 ∈ 𝑥})
6	4, 5	syl 18	. . 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 2864	. 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → 𝑊 ∈ {𝑥 ∈ 𝑇 ∣ 𝐴 ∈ 𝑥})
8		eleq2 2849	. . 3 ⊢ (𝑣 = 𝑊 → (𝐴 ∈ 𝑣 ↔ 𝐴 ∈ 𝑊))
9		eleq2 2849	. . . 4 ⊢ (𝑥 = 𝑣 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑣))
10	9	cbvrabv 3422	. . 3 ⊢ {𝑥 ∈ 𝑇 ∣ 𝐴 ∈ 𝑥} = {𝑣 ∈ 𝑇 ∣ 𝐴 ∈ 𝑣}
11	8, 10	elrab2 3649	. 2 ⊢ (𝑊 ∈ {𝑥 ∈ 𝑇 ∣ 𝐴 ∈ 𝑥} ↔ (𝑊 ∈ 𝑇 ∧ 𝐴 ∈ 𝑊))
12	7, 11	sylib 221	1 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → (𝑊 ∈ 𝑇 ∧ 𝐴 ∈ 𝑊))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3076 ∃!wreu 3363 {crab 3412 ∖ cdif 3896 ∩ cin 3898 ∅c0 4279 𝒫 cpw 4557 {csn 4584 ∪ cuni 4867 ↦ cmpt 5186 ^◡ccnv 5654 ↾ cres 5657 “ cima 5658 ‘cfv 6533 ℩crio 7369 (class class class)co 7413 ↾_t crest 17505 Homeochmeo 23979 CovMap ccvm 35834

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-map 8828 df-top 23119 df-topon 23136 df-cn 23452 df-cvm 35835

This theorem is used by: cvmopnlem 35857 cvmliftmolem2 35861 cvmliftlem6 35869 cvmliftlem8 35871 cvmliftlem9 35872 cvmlift2lem9 35890 cvmlift3lem6 35903 cvmlift3lem7 35904

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