cvmsiota - Mathbox for Mario Carneiro

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Theorem cvmsiota 35299

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 35298	. . . 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 2838	. 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → 𝑊 ∈ {𝑥 ∈ 𝑇 ∣ 𝐴 ∈ 𝑥})
8		eleq2 2823	. . 3 ⊢ (𝑣 = 𝑊 → (𝐴 ∈ 𝑣 ↔ 𝐴 ∈ 𝑊))
9		eleq2 2823	. . . 4 ⊢ (𝑥 = 𝑣 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑣))
10	9	cbvrabv 3426	. . 3 ⊢ {𝑥 ∈ 𝑇 ∣ 𝐴 ∈ 𝑥} = {𝑣 ∈ 𝑇 ∣ 𝐴 ∈ 𝑣}
11	8, 10	elrab2 3674	. 2 ⊢ (𝑊 ∈ {𝑥 ∈ 𝑇 ∣ 𝐴 ∈ 𝑥} ↔ (𝑊 ∈ 𝑇 ∧ 𝐴 ∈ 𝑊))
12	7, 11	sylib 218	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 1086 = wceq 1540 ∈ wcel 2108 ∀wral 3051 ∃!wreu 3357 {crab 3415 ∖ cdif 3923 ∩ cin 3925 ∅c0 4308 𝒫 cpw 4575 {csn 4601 ∪ cuni 4883 ↦ cmpt 5201 ^◡ccnv 5653 ↾ cres 5656 “ cima 5657 ‘cfv 6531 ℩crio 7361 (class class class)co 7405 ↾_t crest 17434 Homeochmeo 23691 CovMap ccvm 35277

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-map 8842 df-top 22832 df-topon 22849 df-cn 23165 df-cvm 35278

This theorem is referenced by: cvmopnlem 35300 cvmliftmolem2 35304 cvmliftlem6 35312 cvmliftlem8 35314 cvmliftlem9 35315 cvmlift2lem9 35333 cvmlift3lem6 35346 cvmlift3lem7 35347

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