elpi1i - Metamath Proof Explorer

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Theorem elpi1i 24973

Description: The elements of the fundamental group. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)

**Hypotheses**
Ref	Expression
elpi1.g	⊢ 𝐺 = (𝐽 π₁ 𝑌)
elpi1.b	⊢ 𝐵 = (Base‘𝐺)
elpi1.1	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
elpi1.2	⊢ (𝜑 → 𝑌 ∈ 𝑋)
elpi1i.3	⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
elpi1i.4	⊢ (𝜑 → (𝐹‘0) = 𝑌)
elpi1i.5	⊢ (𝜑 → (𝐹‘1) = 𝑌)

**Assertion**
Ref	Expression
elpi1i	⊢ (𝜑 → [𝐹]( ≃_ph‘𝐽) ∈ 𝐵)

Proof of Theorem elpi1i Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpi1i.3	. . 3 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
2		elpi1i.4	. . 3 ⊢ (𝜑 → (𝐹‘0) = 𝑌)
3		elpi1i.5	. . 3 ⊢ (𝜑 → (𝐹‘1) = 𝑌)
4		eceq1 8661	. . . . . . 7 ⊢ (𝑓 = 𝐹 → [𝑓]( ≃_ph‘𝐽) = [𝐹]( ≃_ph‘𝐽))
5	4	eqcomd 2737	. . . . . 6 ⊢ (𝑓 = 𝐹 → [𝐹]( ≃_ph‘𝐽) = [𝑓]( ≃_ph‘𝐽))
6	5	biantrud 531	. . . . 5 ⊢ (𝑓 = 𝐹 → (((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌) ↔ (((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌) ∧ [𝐹]( ≃_ph‘𝐽) = [𝑓]( ≃_ph‘𝐽))))
7		fveq1 6821	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘0) = (𝐹‘0))
8	7	eqeq1d 2733	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑓‘0) = 𝑌 ↔ (𝐹‘0) = 𝑌))
9		fveq1 6821	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘1) = (𝐹‘1))
10	9	eqeq1d 2733	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑓‘1) = 𝑌 ↔ (𝐹‘1) = 𝑌))
11	8, 10	anbi12d 632	. . . . 5 ⊢ (𝑓 = 𝐹 → (((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌) ↔ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑌)))
12	6, 11	bitr3d 281	. . . 4 ⊢ (𝑓 = 𝐹 → ((((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌) ∧ [𝐹]( ≃_ph‘𝐽) = [𝑓]( ≃_ph‘𝐽)) ↔ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑌)))
13	12	rspcev 3572	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑌)) → ∃𝑓 ∈ (II Cn 𝐽)(((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌) ∧ [𝐹]( ≃_ph‘𝐽) = [𝑓]( ≃_ph‘𝐽)))
14	1, 2, 3, 13	syl12anc 836	. 2 ⊢ (𝜑 → ∃𝑓 ∈ (II Cn 𝐽)(((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌) ∧ [𝐹]( ≃_ph‘𝐽) = [𝑓]( ≃_ph‘𝐽)))
15		elpi1.g	. . 3 ⊢ 𝐺 = (𝐽 π₁ 𝑌)
16		elpi1.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
17		elpi1.1	. . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
18		elpi1.2	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑋)
19	15, 16, 17, 18	elpi1 24972	. 2 ⊢ (𝜑 → ([𝐹]( ≃_ph‘𝐽) ∈ 𝐵 ↔ ∃𝑓 ∈ (II Cn 𝐽)(((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌) ∧ [𝐹]( ≃_ph‘𝐽) = [𝑓]( ≃_ph‘𝐽))))
20	14, 19	mpbird 257	1 ⊢ (𝜑 → [𝐹]( ≃_ph‘𝐽) ∈ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ∃wrex 3056 ‘cfv 6481 (class class class)co 7346 [cec 8620 0cc0 11006 1c1 11007 Basecbs 17120 TopOnctopon 22825 Cn ccn 23139 IIcii 24795 ≃_phcphtpc 24895 π₁ cpi1 24930

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-ec 8624 df-qs 8628 df-map 8752 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-icc 13252 df-fz 13408 df-fzo 13555 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-qus 17413 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-mulg 18981 df-cntz 19229 df-cmn 19694 df-psmet 21283 df-xmet 21284 df-met 21285 df-bl 21286 df-mopn 21287 df-cnfld 21292 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22861 df-cld 22934 df-cn 23142 df-cnp 23143 df-tx 23477 df-hmeo 23670 df-xms 24235 df-ms 24236 df-tms 24237 df-ii 24797 df-htpy 24896 df-phtpy 24897 df-phtpc 24918 df-om1 24933 df-pi1 24935

This theorem is referenced by: pi1inv 24979 pi1xfrf 24980 pi1cof 24986 sconnpi1 35283

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