cvmliftiota - Mathbox for Mario Carneiro

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Theorem cvmliftiota 32548

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 5728	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝐹 ∘ 𝑓) = (𝐹 ∘ 𝑔))
3	2	eqeq1d 2823	. . . . . 6 ⊢ (𝑓 = 𝑔 → ((𝐹 ∘ 𝑓) = 𝐺 ↔ (𝐹 ∘ 𝑔) = 𝐺))
4		fveq1 6668	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑓‘0) = (𝑔‘0))
5	4	eqeq1d 2823	. . . . . 6 ⊢ (𝑓 = 𝑔 → ((𝑓‘0) = 𝑃 ↔ (𝑔‘0) = 𝑃))
6	3, 5	anbi12d 632	. . . . 5 ⊢ (𝑓 = 𝑔 → (((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘0) = 𝑃)))
7	6	cbvriotavw 7123	. . . 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 2844	. . 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 32546	. . . . 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 836	. . . 4 ⊢ (𝜑 → ∃!𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘0) = 𝑃))
16		riotacl2 7129	. . . 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 17	. . 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 2917	. 2 ⊢ (𝜑 → 𝐻 ∈ {𝑔 ∈ (II Cn 𝐶) ∣ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘0) = 𝑃)})
19		coeq2 5728	. . . . . 6 ⊢ (𝑔 = 𝐻 → (𝐹 ∘ 𝑔) = (𝐹 ∘ 𝐻))
20	19	eqeq1d 2823	. . . . 5 ⊢ (𝑔 = 𝐻 → ((𝐹 ∘ 𝑔) = 𝐺 ↔ (𝐹 ∘ 𝐻) = 𝐺))
21		fveq1 6668	. . . . . 6 ⊢ (𝑔 = 𝐻 → (𝑔‘0) = (𝐻‘0))
22	21	eqeq1d 2823	. . . . 5 ⊢ (𝑔 = 𝐻 → ((𝑔‘0) = 𝑃 ↔ (𝐻‘0) = 𝑃))
23	20, 22	anbi12d 632	. . . 4 ⊢ (𝑔 = 𝐻 → (((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹 ∘ 𝐻) = 𝐺 ∧ (𝐻‘0) = 𝑃)))
24	23	elrab 3679	. . 3 ⊢ (𝐻 ∈ {𝑔 ∈ (II Cn 𝐶) ∣ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘0) = 𝑃)} ↔ (𝐻 ∈ (II Cn 𝐶) ∧ ((𝐹 ∘ 𝐻) = 𝐺 ∧ (𝐻‘0) = 𝑃)))
25		3anass 1091	. . 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 280	. 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 220	1 ⊢ (𝜑 → (𝐻 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐻) = 𝐺 ∧ (𝐻‘0) = 𝑃))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∃!wreu 3140 {crab 3142 ∪ cuni 4837 ∘ ccom 5558 ‘cfv 6354 ℩crio 7112 (class class class)co 7155 0cc0 10536 Cn ccn 21831 IIcii 23482 CovMap ccvm 32502

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-addf 10615 ax-mulf 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-ec 8290 df-map 8407 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-fi 8874 df-sup 8905 df-inf 8906 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-q 12348 df-rp 12389 df-xneg 12506 df-xadd 12507 df-xmul 12508 df-ioo 12741 df-ico 12743 df-icc 12744 df-fz 12892 df-fzo 13033 df-fl 13161 df-seq 13369 df-exp 13429 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-clim 14844 df-sum 15042 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-hom 16588 df-cco 16589 df-rest 16695 df-topn 16696 df-0g 16714 df-gsum 16715 df-topgen 16716 df-pt 16717 df-prds 16720 df-xrs 16774 df-qtop 16779 df-imas 16780 df-xps 16782 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-submnd 17956 df-mulg 18224 df-cntz 18446 df-cmn 18907 df-psmet 20536 df-xmet 20537 df-met 20538 df-bl 20539 df-mopn 20540 df-cnfld 20545 df-top 21501 df-topon 21518 df-topsp 21540 df-bases 21553 df-cld 21626 df-ntr 21627 df-cls 21628 df-nei 21705 df-cn 21834 df-cnp 21835 df-cmp 21994 df-conn 22019 df-lly 22073 df-nlly 22074 df-tx 22169 df-hmeo 22362 df-xms 22929 df-ms 22930 df-tms 22931 df-ii 23484 df-htpy 23573 df-phtpy 23574 df-phtpc 23595 df-pconn 32468 df-sconn 32469 df-cvm 32503

This theorem is referenced by: cvmlift2lem2 32551 cvmlift2lem3 32552 cvmliftphtlem 32564 cvmliftpht 32565 cvmlift3lem2 32567 cvmlift3lem4 32569 cvmlift3lem5 32570 cvmlift3lem6 32571

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