elpi1i - Metamath Proof Explorer

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Theorem elpi1i 24258

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 8567	. . . . . . 7 ⊢ (𝑓 = 𝐹 → [𝑓]( ≃_ph‘𝐽) = [𝐹]( ≃_ph‘𝐽))
5	4	eqcomd 2742	. . . . . 6 ⊢ (𝑓 = 𝐹 → [𝐹]( ≃_ph‘𝐽) = [𝑓]( ≃_ph‘𝐽))
6	5	biantrud 533	. . . . 5 ⊢ (𝑓 = 𝐹 → (((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌) ↔ (((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌) ∧ [𝐹]( ≃_ph‘𝐽) = [𝑓]( ≃_ph‘𝐽))))
7		fveq1 6803	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘0) = (𝐹‘0))
8	7	eqeq1d 2738	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑓‘0) = 𝑌 ↔ (𝐹‘0) = 𝑌))
9		fveq1 6803	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘1) = (𝐹‘1))
10	9	eqeq1d 2738	. . . . . 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 3566	. . 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 835	. 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 24257	. 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 397 = wceq 1539 ∈ wcel 2104 ∃wrex 3071 ‘cfv 6458 (class class class)co 7307 [cec 8527 0cc0 10921 1c1 10922 Basecbs 16961 TopOnctopon 22108 Cn ccn 22424 IIcii 24087 ≃_phcphtpc 24181 π₁ cpi1 24215

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10977 ax-resscn 10978 ax-1cn 10979 ax-icn 10980 ax-addcl 10981 ax-addrcl 10982 ax-mulcl 10983 ax-mulrcl 10984 ax-mulcom 10985 ax-addass 10986 ax-mulass 10987 ax-distr 10988 ax-i2m1 10989 ax-1ne0 10990 ax-1rid 10991 ax-rnegex 10992 ax-rrecex 10993 ax-cnre 10994 ax-pre-lttri 10995 ax-pre-lttrn 10996 ax-pre-ltadd 10997 ax-pre-mulgt0 10998 ax-pre-sup 10999 ax-mulf 11001

This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3304 df-reu 3305 df-rab 3306 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-tp 4570 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-iin 4934 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-se 5556 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-isom 6467 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-of 7565 df-om 7745 df-1st 7863 df-2nd 7864 df-supp 8009 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-2o 8329 df-er 8529 df-ec 8531 df-qs 8535 df-map 8648 df-ixp 8717 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-fsupp 9177 df-fi 9218 df-sup 9249 df-inf 9250 df-oi 9317 df-card 9745 df-pnf 11061 df-mnf 11062 df-xr 11063 df-ltxr 11064 df-le 11065 df-sub 11257 df-neg 11258 df-div 11683 df-nn 12024 df-2 12086 df-3 12087 df-4 12088 df-5 12089 df-6 12090 df-7 12091 df-8 12092 df-9 12093 df-n0 12284 df-z 12370 df-dec 12488 df-uz 12633 df-q 12739 df-rp 12781 df-xneg 12898 df-xadd 12899 df-xmul 12900 df-ioo 13133 df-icc 13136 df-fz 13290 df-fzo 13433 df-seq 13772 df-exp 13833 df-hash 14095 df-cj 14859 df-re 14860 df-im 14861 df-sqrt 14995 df-abs 14996 df-struct 16897 df-sets 16914 df-slot 16932 df-ndx 16944 df-base 16962 df-ress 16991 df-plusg 17024 df-mulr 17025 df-starv 17026 df-sca 17027 df-vsca 17028 df-ip 17029 df-tset 17030 df-ple 17031 df-ds 17033 df-unif 17034 df-hom 17035 df-cco 17036 df-rest 17182 df-topn 17183 df-0g 17201 df-gsum 17202 df-topgen 17203 df-pt 17204 df-prds 17207 df-xrs 17262 df-qtop 17267 df-imas 17268 df-qus 17269 df-xps 17270 df-mre 17344 df-mrc 17345 df-acs 17347 df-mgm 18375 df-sgrp 18424 df-mnd 18435 df-submnd 18480 df-mulg 18750 df-cntz 18972 df-cmn 19437 df-psmet 20638 df-xmet 20639 df-met 20640 df-bl 20641 df-mopn 20642 df-cnfld 20647 df-top 22092 df-topon 22109 df-topsp 22131 df-bases 22145 df-cld 22219 df-cn 22427 df-cnp 22428 df-tx 22762 df-hmeo 22955 df-xms 23522 df-ms 23523 df-tms 23524 df-ii 24089 df-htpy 24182 df-phtpy 24183 df-phtpc 24204 df-om1 24218 df-pi1 24220

This theorem is referenced by: pi1inv 24264 pi1xfrf 24265 pi1cof 24271 sconnpi1 33250

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