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Mirrors > Home > MPE Home > Th. List > brcici | Structured version Visualization version GIF version |
Description: Prove that two objects are isomorphic by an explicit isomorphism. (Contributed by AV, 4-Apr-2020.) |
Ref | Expression |
---|---|
cic.i | ⊢ 𝐼 = (Iso‘𝐶) |
cic.b | ⊢ 𝐵 = (Base‘𝐶) |
cic.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
cic.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
cic.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
cic.f | ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) |
Ref | Expression |
---|---|
brcici | ⊢ (𝜑 → 𝑋( ≃𝑐 ‘𝐶)𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cic.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) | |
2 | eleq1 2824 | . . . 4 ⊢ (𝑓 = 𝐹 → (𝑓 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋𝐼𝑌))) | |
3 | 2 | spcegv 3531 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐼𝑌) → (𝐹 ∈ (𝑋𝐼𝑌) → ∃𝑓 𝑓 ∈ (𝑋𝐼𝑌))) |
4 | 1, 1, 3 | sylc 65 | . 2 ⊢ (𝜑 → ∃𝑓 𝑓 ∈ (𝑋𝐼𝑌)) |
5 | cic.i | . . 3 ⊢ 𝐼 = (Iso‘𝐶) | |
6 | cic.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
7 | cic.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
8 | cic.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
9 | cic.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
10 | 5, 6, 7, 8, 9 | cic 17455 | . 2 ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ ∃𝑓 𝑓 ∈ (𝑋𝐼𝑌))) |
11 | 4, 10 | mpbird 256 | 1 ⊢ (𝜑 → 𝑋( ≃𝑐 ‘𝐶)𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∃wex 1783 ∈ wcel 2107 class class class wbr 5075 ‘cfv 6423 (class class class)co 7260 Basecbs 16856 Catccat 17317 Isociso 17402 ≃𝑐 ccic 17451 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5210 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7571 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3429 df-sbc 3717 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-iun 4928 df-br 5076 df-opab 5138 df-mpt 5159 df-id 5485 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-iota 6381 df-fun 6425 df-fn 6426 df-f 6427 df-f1 6428 df-fo 6429 df-f1o 6430 df-fv 6431 df-ov 7263 df-oprab 7264 df-mpo 7265 df-1st 7809 df-2nd 7810 df-supp 7954 df-inv 17404 df-iso 17405 df-cic 17452 |
This theorem is referenced by: cicref 17457 cicsym 17460 cictr 17461 |
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