cvmliftiota - Mathbox for Mario Carneiro

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Theorem cvmliftiota 36246

Description: Write out a function 𝐻 that is the unique lift of 𝐹. (Contributed by Mario Carneiro, 16-Feb-2015.)

**Hypotheses**
Ref	Expression
cvmliftiota.b	⊢ 𝐵 = ∪ 𝐶
cvmliftiota.h	⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))
cvmliftiota.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
cvmliftiota.g	⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))
cvmliftiota.p	⊢ (𝜑 → 𝑃 ∈ 𝐵)
cvmliftiota.e	⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))

**Assertion**
Ref	Expression
cvmliftiota	⊢ (𝜑 → (𝐻 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐻) = 𝐺 ∧ (𝐻‘0) = 𝑃))

Distinct variable groups: 𝐶,𝑓 𝑓,𝐹 𝑓,𝐺 𝑃,𝑓 Allowed substitution hints: 𝜑(𝑓) 𝐵(𝑓) 𝐻(𝑓) 𝐽(𝑓)

Proof of Theorem cvmliftiota Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvmliftiota.h	. . . 4 ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))
2		coeq2 5843	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝐹 ∘ 𝑓) = (𝐹 ∘ 𝑔))
3	2	eqeq1d 2765	. . . . . 6 ⊢ (𝑓 = 𝑔 → ((𝐹 ∘ 𝑓) = 𝐺 ↔ (𝐹 ∘ 𝑔) = 𝐺))
4		fveq1 6910	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑓‘0) = (𝑔‘0))
5	4	eqeq1d 2765	. . . . . 6 ⊢ (𝑓 = 𝑔 → ((𝑓‘0) = 𝑃 ↔ (𝑔‘0) = 𝑃))
6	3, 5	anbi12d 644	. . . . 5 ⊢ (𝑓 = 𝑔 → (((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘0) = 𝑃)))
7	6	cbvriotavw 7413	. . . 4 ⊢ (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)) = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘0) = 𝑃))
8	1, 7	eqtri 2786	. . 3 ⊢ 𝐻 = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘0) = 𝑃))
9		cvmliftiota.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
10		cvmliftiota.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))
11		cvmliftiota.p	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
12		cvmliftiota.e	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))
13		cvmliftiota.b	. . . . . 6 ⊢ 𝐵 = ∪ 𝐶
14	13	cvmlift 36244	. . . . 5 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽)) ∧ (𝑃 ∈ 𝐵 ∧ (𝐹‘𝑃) = (𝐺‘0))) → ∃!𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘0) = 𝑃))
15	9, 10, 11, 12, 14	syl22anc 852	. . . 4 ⊢ (𝜑 → ∃!𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘0) = 𝑃))
16		riotacl2 7419	. . . 4 ⊢ (∃!𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘0) = 𝑃) → (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘0) = 𝑃)) ∈ {𝑔 ∈ (II Cn 𝐶) ∣ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘0) = 𝑃)})
17	15, 16	syl 18	. . 3 ⊢ (𝜑 → (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘0) = 𝑃)) ∈ {𝑔 ∈ (II Cn 𝐶) ∣ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘0) = 𝑃)})
18	8, 17	eqeltrid 2868	. 2 ⊢ (𝜑 → 𝐻 ∈ {𝑔 ∈ (II Cn 𝐶) ∣ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘0) = 𝑃)})
19		coeq2 5843	. . . . . 6 ⊢ (𝑔 = 𝐻 → (𝐹 ∘ 𝑔) = (𝐹 ∘ 𝐻))
20	19	eqeq1d 2765	. . . . 5 ⊢ (𝑔 = 𝐻 → ((𝐹 ∘ 𝑔) = 𝐺 ↔ (𝐹 ∘ 𝐻) = 𝐺))
21		fveq1 6910	. . . . . 6 ⊢ (𝑔 = 𝐻 → (𝑔‘0) = (𝐻‘0))
22	21	eqeq1d 2765	. . . . 5 ⊢ (𝑔 = 𝐻 → ((𝑔‘0) = 𝑃 ↔ (𝐻‘0) = 𝑃))
23	20, 22	anbi12d 644	. . . 4 ⊢ (𝑔 = 𝐻 → (((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹 ∘ 𝐻) = 𝐺 ∧ (𝐻‘0) = 𝑃)))
24	23	elrab 3648	. . 3 ⊢ (𝐻 ∈ {𝑔 ∈ (II Cn 𝐶) ∣ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘0) = 𝑃)} ↔ (𝐻 ∈ (II Cn 𝐶) ∧ ((𝐹 ∘ 𝐻) = 𝐺 ∧ (𝐻‘0) = 𝑃)))
25		3anass 1111	. . 3 ⊢ ((𝐻 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐻) = 𝐺 ∧ (𝐻‘0) = 𝑃) ↔ (𝐻 ∈ (II Cn 𝐶) ∧ ((𝐹 ∘ 𝐻) = 𝐺 ∧ (𝐻‘0) = 𝑃)))
26	24, 25	bitr4i 281	. 2 ⊢ (𝐻 ∈ {𝑔 ∈ (II Cn 𝐶) ∣ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘0) = 𝑃)} ↔ (𝐻 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐻) = 𝐺 ∧ (𝐻‘0) = 𝑃))
27	18, 26	sylib 221	1 ⊢ (𝜑 → (𝐻 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐻) = 𝐺 ∧ (𝐻‘0) = 𝑃))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ∃!wreu 3367 {crab 3416 ∪ cuni 4870 ∘ ccom 5660 ‘cfv 6561 ℩crio 7402 (class class class)co 7446 0cc0 11230 Cn ccn 23575 IIcii 25229 CovMap ccvm 36200

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778 ax-inf2 9672 ax-cnex 11286 ax-resscn 11287 ax-1cn 11288 ax-icn 11289 ax-addcl 11290 ax-addrcl 11291 ax-mulcl 11292 ax-mulrcl 11293 ax-mulcom 11294 ax-addass 11295 ax-mulass 11296 ax-distr 11297 ax-i2m1 11298 ax-1ne0 11299 ax-1rid 11300 ax-rnegex 11301 ax-rrecex 11302 ax-cnre 11303 ax-pre-lttri 11304 ax-pre-lttrn 11305 ax-pre-ltadd 11306 ax-pre-mulgt0 11307 ax-pre-sup 11308 ax-addf 11309

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-se 5610 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6325 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7403 df-ov 7449 df-oprab 7450 df-mpo 7451 df-of 7720 df-om 7905 df-1st 8028 df-2nd 8029 df-supp 8208 df-frecs 8330 df-wrecs 8361 df-recs 8410 df-rdg 8449 df-1o 8507 df-2o 8508 df-er 8748 df-ec 8750 df-map 8880 df-ixp 8957 df-en 9005 df-dom 9006 df-sdom 9007 df-fin 9008 df-fsupp 9385 df-fi 9433 df-sup 9464 df-inf 9465 df-oi 9534 df-card 10050 df-pnf 11375 df-mnf 11376 df-xr 11377 df-ltxr 11378 df-le 11379 df-sub 11573 df-neg 11574 df-div 12004 df-nn 12366 df-2 12435 df-3 12436 df-4 12437 df-5 12438 df-6 12439 df-7 12440 df-8 12441 df-9 12442 df-n0 12637 df-z 12724 df-dec 12845 df-uz 12996 df-q 13106 df-rp 13151 df-xneg 13271 df-xadd 13272 df-xmul 13273 df-ioo 13510 df-ico 13512 df-icc 13513 df-fz 13670 df-fzo 13819 df-fl 13962 df-seq 14175 df-exp 14235 df-hash 14505 df-cj 15298 df-re 15299 df-im 15300 df-sqrt 15434 df-abs 15435 df-clim 15687 df-sum 15886 df-struct 17357 df-sets 17374 df-slot 17392 df-ndx 17404 df-base 17420 df-ress 17441 df-plusg 17473 df-mulr 17474 df-starv 17475 df-sca 17476 df-vsca 17477 df-ip 17478 df-tset 17479 df-ple 17480 df-ds 17482 df-unif 17483 df-hom 17484 df-cco 17485 df-rest 17625 df-topn 17626 df-0g 17644 df-gsum 17645 df-topgen 17646 df-pt 17647 df-prds 17650 df-xrs 17706 df-qtop 17711 df-imas 17712 df-xps 17714 df-mre 17788 df-mrc 17789 df-acs 17791 df-mgm 18848 df-sgrp 18940 df-mnd 18956 df-submnd 19011 df-mulg 19310 df-cntz 19564 df-cmn 20029 df-psmet 21703 df-xmet 21704 df-met 21705 df-bl 21706 df-mopn 21707 df-cnfld 21712 df-top 23245 df-topon 23262 df-topsp 23284 df-bases 23297 df-cld 23370 df-ntr 23371 df-cls 23372 df-nei 23449 df-cn 23578 df-cnp 23579 df-cmp 23738 df-conn 23763 df-lly 23818 df-nlly 23819 df-tx 23914 df-hmeo 24107 df-xms 24672 df-ms 24673 df-tms 24674 df-ii 25231 df-cncf 25232 df-htpy 25324 df-phtpy 25325 df-phtpc 25346 df-pconn 36166 df-sconn 36167 df-cvm 36201

This theorem is used by: cvmlift2lem2 36249 cvmlift2lem3 36250 cvmliftphtlem 36262 cvmliftpht 36263 cvmlift3lem2 36265 cvmlift3lem4 36267 cvmlift3lem5 36268 cvmlift3lem6 36269

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