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Mathbox for Mario Carneiro |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmliftiota | Structured version Visualization version GIF version |
Description: Write out a function 𝐻 that is the unique lift of 𝐹. (Contributed by Mario Carneiro, 16-Feb-2015.) |
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)) |
Ref | Expression |
---|---|
cvmliftiota | ⊢ (𝜑 → (𝐻 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐻) = 𝐺 ∧ (𝐻‘0) = 𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvmliftiota.h | . . . 4 ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)) | |
2 | coeq2 5858 | . . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝐹 ∘ 𝑓) = (𝐹 ∘ 𝑔)) | |
3 | 2 | eqeq1d 2733 | . . . . . 6 ⊢ (𝑓 = 𝑔 → ((𝐹 ∘ 𝑓) = 𝐺 ↔ (𝐹 ∘ 𝑔) = 𝐺)) |
4 | fveq1 6890 | . . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑓‘0) = (𝑔‘0)) | |
5 | 4 | eqeq1d 2733 | . . . . . 6 ⊢ (𝑓 = 𝑔 → ((𝑓‘0) = 𝑃 ↔ (𝑔‘0) = 𝑃)) |
6 | 3, 5 | anbi12d 630 | . . . . 5 ⊢ (𝑓 = 𝑔 → (((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘0) = 𝑃))) |
7 | 6 | cbvriotavw 7378 | . . . 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 2759 | . . 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 34603 | . . . . 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 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 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 2836 | . 2 ⊢ (𝜑 → 𝐻 ∈ {𝑔 ∈ (II Cn 𝐶) ∣ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘0) = 𝑃)}) |
19 | coeq2 5858 | . . . . . 6 ⊢ (𝑔 = 𝐻 → (𝐹 ∘ 𝑔) = (𝐹 ∘ 𝐻)) | |
20 | 19 | eqeq1d 2733 | . . . . 5 ⊢ (𝑔 = 𝐻 → ((𝐹 ∘ 𝑔) = 𝐺 ↔ (𝐹 ∘ 𝐻) = 𝐺)) |
21 | fveq1 6890 | . . . . . 6 ⊢ (𝑔 = 𝐻 → (𝑔‘0) = (𝐻‘0)) | |
22 | 21 | eqeq1d 2733 | . . . . 5 ⊢ (𝑔 = 𝐻 → ((𝑔‘0) = 𝑃 ↔ (𝐻‘0) = 𝑃)) |
23 | 20, 22 | anbi12d 630 | . . . 4 ⊢ (𝑔 = 𝐻 → (((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹 ∘ 𝐻) = 𝐺 ∧ (𝐻‘0) = 𝑃))) |
24 | 23 | elrab 3683 | . . 3 ⊢ (𝐻 ∈ {𝑔 ∈ (II Cn 𝐶) ∣ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘0) = 𝑃)} ↔ (𝐻 ∈ (II Cn 𝐶) ∧ ((𝐹 ∘ 𝐻) = 𝐺 ∧ (𝐻‘0) = 𝑃))) |
25 | 3anass 1094 | . . 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 278 | . 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 217 | 1 ⊢ (𝜑 → (𝐻 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐻) = 𝐺 ∧ (𝐻‘0) = 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ∃!wreu 3373 {crab 3431 ∪ cuni 4908 ∘ ccom 5680 ‘cfv 6543 ℩crio 7367 (class class class)co 7412 0cc0 11116 Cn ccn 22961 IIcii 24628 CovMap ccvm 34559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-addf 11195 ax-mulf 11196 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-er 8709 df-ec 8711 df-map 8828 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-fi 9412 df-sup 9443 df-inf 9444 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-q 12940 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-ioo 13335 df-ico 13337 df-icc 13338 df-fz 13492 df-fzo 13635 df-fl 13764 df-seq 13974 df-exp 14035 df-hash 14298 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15439 df-sum 15640 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-pt 17397 df-prds 17400 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18568 df-sgrp 18647 df-mnd 18663 df-submnd 18709 df-mulg 18991 df-cntz 19226 df-cmn 19695 df-psmet 21140 df-xmet 21141 df-met 21142 df-bl 21143 df-mopn 21144 df-cnfld 21149 df-top 22629 df-topon 22646 df-topsp 22668 df-bases 22682 df-cld 22756 df-ntr 22757 df-cls 22758 df-nei 22835 df-cn 22964 df-cnp 22965 df-cmp 23124 df-conn 23149 df-lly 23203 df-nlly 23204 df-tx 23299 df-hmeo 23492 df-xms 24059 df-ms 24060 df-tms 24061 df-ii 24630 df-htpy 24729 df-phtpy 24730 df-phtpc 24751 df-pconn 34525 df-sconn 34526 df-cvm 34560 |
This theorem is referenced by: cvmlift2lem2 34608 cvmlift2lem3 34609 cvmliftphtlem 34621 cvmliftpht 34622 cvmlift3lem2 34624 cvmlift3lem4 34626 cvmlift3lem5 34627 cvmlift3lem6 34628 |
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