cvmliftiota - Mathbox for Mario Carneiro

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Theorem cvmliftiota 36035

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 5836	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝐹 ∘ 𝑓) = (𝐹 ∘ 𝑔))
3	2	eqeq1d 2763	. . . . . 6 ⊢ (𝑓 = 𝑔 → ((𝐹 ∘ 𝑓) = 𝐺 ↔ (𝐹 ∘ 𝑔) = 𝐺))
4		fveq1 6876	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑓‘0) = (𝑔‘0))
5	4	eqeq1d 2763	. . . . . 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 7379	. . . 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 2784	. . 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 36033	. . . . 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 7385	. . . 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 2865	. 2 ⊢ (𝜑 → 𝐻 ∈ {𝑔 ∈ (II Cn 𝐶) ∣ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘0) = 𝑃)})
19		coeq2 5836	. . . . . 6 ⊢ (𝑔 = 𝐻 → (𝐹 ∘ 𝑔) = (𝐹 ∘ 𝐻))
20	19	eqeq1d 2763	. . . . 5 ⊢ (𝑔 = 𝐻 → ((𝐹 ∘ 𝑔) = 𝐺 ↔ (𝐹 ∘ 𝐻) = 𝐺))
21		fveq1 6876	. . . . . 6 ⊢ (𝑔 = 𝐻 → (𝑔‘0) = (𝐻‘0))
22	21	eqeq1d 2763	. . . . 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 3645	. . 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 2145 ∃!wreu 3364 {crab 3413 ∪ cuni 4867 ∘ ccom 5655 ‘cfv 6531 ℩crio 7368 (class class class)co 7412 0cc0 11181 Cn ccn 23522 IIcii 25176 CovMap ccvm 35989

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-inf2 9626 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258 ax-pre-sup 11259 ax-addf 11260

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 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-2o 8461 df-er 8701 df-ec 8703 df-map 8833 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9338 df-fi 9387 df-sup 9418 df-inf 9419 df-oi 9488 df-card 10001 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-div 11955 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-7 12391 df-8 12392 df-9 12393 df-n0 12588 df-z 12675 df-dec 12796 df-uz 12947 df-q 13057 df-rp 13102 df-xneg 13222 df-xadd 13223 df-xmul 13224 df-ioo 13461 df-ico 13463 df-icc 13464 df-fz 13621 df-fzo 13769 df-fl 13912 df-seq 14125 df-exp 14185 df-hash 14455 df-cj 15246 df-re 15247 df-im 15248 df-sqrt 15382 df-abs 15383 df-clim 15635 df-sum 15834 df-struct 17305 df-sets 17322 df-slot 17340 df-ndx 17352 df-base 17368 df-ress 17389 df-plusg 17421 df-mulr 17422 df-starv 17423 df-sca 17424 df-vsca 17425 df-ip 17426 df-tset 17427 df-ple 17428 df-ds 17430 df-unif 17431 df-hom 17432 df-cco 17433 df-rest 17573 df-topn 17574 df-0g 17592 df-gsum 17593 df-topgen 17594 df-pt 17595 df-prds 17598 df-xrs 17654 df-qtop 17659 df-imas 17660 df-xps 17662 df-mre 17736 df-mrc 17737 df-acs 17739 df-mgm 18796 df-sgrp 18888 df-mnd 18904 df-submnd 18959 df-mulg 19258 df-cntz 19511 df-cmn 19976 df-psmet 21650 df-xmet 21651 df-met 21652 df-bl 21653 df-mopn 21654 df-cnfld 21659 df-top 23192 df-topon 23209 df-topsp 23231 df-bases 23244 df-cld 23317 df-ntr 23318 df-cls 23319 df-nei 23396 df-cn 23525 df-cnp 23526 df-cmp 23685 df-conn 23710 df-lly 23765 df-nlly 23766 df-tx 23861 df-hmeo 24054 df-xms 24619 df-ms 24620 df-tms 24621 df-ii 25178 df-cncf 25179 df-htpy 25271 df-phtpy 25272 df-phtpc 25293 df-pconn 35955 df-sconn 35956 df-cvm 35990

This theorem is used by: cvmlift2lem2 36038 cvmlift2lem3 36039 cvmliftphtlem 36051 cvmliftpht 36052 cvmlift3lem2 36054 cvmlift3lem4 36056 cvmlift3lem5 36057 cvmlift3lem6 36058

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