elpi1i - Metamath Proof Explorer

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Theorem elpi1i 24190

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 8510	. . . . . . 7 ⊢ (𝑓 = 𝐹 → [𝑓]( ≃_ph‘𝐽) = [𝐹]( ≃_ph‘𝐽))
5	4	eqcomd 2745	. . . . . 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 6767	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘0) = (𝐹‘0))
8	7	eqeq1d 2741	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑓‘0) = 𝑌 ↔ (𝐹‘0) = 𝑌))
9		fveq1 6767	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘1) = (𝐹‘1))
10	9	eqeq1d 2741	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑓‘1) = 𝑌 ↔ (𝐹‘1) = 𝑌))
11	8, 10	anbi12d 630	. . . . 5 ⊢ (𝑓 = 𝐹 → (((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌) ↔ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑌)))
12	6, 11	bitr3d 280	. . . 4 ⊢ (𝑓 = 𝐹 → ((((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌) ∧ [𝐹]( ≃_ph‘𝐽) = [𝑓]( ≃_ph‘𝐽)) ↔ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑌)))
13	12	rspcev 3560	. . 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 833	. 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 24189	. 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 256	1 ⊢ (𝜑 → [𝐹]( ≃_ph‘𝐽) ∈ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2109 ∃wrex 3066 ‘cfv 6430 (class class class)co 7268 [cec 8470 0cc0 10855 1c1 10856 Basecbs 16893 TopOnctopon 22040 Cn ccn 22356 IIcii 24019 ≃_phcphtpc 24113 π₁ cpi1 24147

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-pre-sup 10933 ax-mulf 10935

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-iin 4932 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-se 5544 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-isom 6439 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-om 7701 df-1st 7817 df-2nd 7818 df-supp 7962 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-2o 8282 df-er 8472 df-ec 8474 df-qs 8478 df-map 8591 df-ixp 8660 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-fsupp 9090 df-fi 9131 df-sup 9162 df-inf 9163 df-oi 9230 df-card 9681 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-7 12024 df-8 12025 df-9 12026 df-n0 12217 df-z 12303 df-dec 12420 df-uz 12565 df-q 12671 df-rp 12713 df-xneg 12830 df-xadd 12831 df-xmul 12832 df-ioo 13065 df-icc 13068 df-fz 13222 df-fzo 13365 df-seq 13703 df-exp 13764 df-hash 14026 df-cj 14791 df-re 14792 df-im 14793 df-sqrt 14927 df-abs 14928 df-struct 16829 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-mulr 16957 df-starv 16958 df-sca 16959 df-vsca 16960 df-ip 16961 df-tset 16962 df-ple 16963 df-ds 16965 df-unif 16966 df-hom 16967 df-cco 16968 df-rest 17114 df-topn 17115 df-0g 17133 df-gsum 17134 df-topgen 17135 df-pt 17136 df-prds 17139 df-xrs 17194 df-qtop 17199 df-imas 17200 df-qus 17201 df-xps 17202 df-mre 17276 df-mrc 17277 df-acs 17279 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-submnd 18412 df-mulg 18682 df-cntz 18904 df-cmn 19369 df-psmet 20570 df-xmet 20571 df-met 20572 df-bl 20573 df-mopn 20574 df-cnfld 20579 df-top 22024 df-topon 22041 df-topsp 22063 df-bases 22077 df-cld 22151 df-cn 22359 df-cnp 22360 df-tx 22694 df-hmeo 22887 df-xms 23454 df-ms 23455 df-tms 23456 df-ii 24021 df-htpy 24114 df-phtpy 24115 df-phtpc 24136 df-om1 24150 df-pi1 24152

This theorem is referenced by: pi1inv 24196 pi1xfrf 24197 pi1cof 24203 sconnpi1 33180

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