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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 2676 | . . . 4 ⊢ (𝑓 = 𝐹 → (𝑓 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋𝐼𝑌))) | |
3 | 2 | spcegv 3267 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐼𝑌) → (𝐹 ∈ (𝑋𝐼𝑌) → ∃𝑓 𝑓 ∈ (𝑋𝐼𝑌))) |
4 | 1, 1, 3 | sylc 63 | . 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 16282 | . 2 ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ ∃𝑓 𝑓 ∈ (𝑋𝐼𝑌))) |
11 | 4, 10 | mpbird 246 | 1 ⊢ (𝜑 → 𝑋( ≃𝑐 ‘𝐶)𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∃wex 1695 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Catccat 16148 Isociso 16229 ≃𝑐 ccic 16278 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-supp 7183 df-inv 16231 df-iso 16232 df-cic 16279 |
This theorem is referenced by: cicref 16284 cicsym 16287 cictr 16288 |
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