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Mirrors > Home > MPE Home > Th. List > invinv | Structured version Visualization version GIF version |
Description: The inverse of the inverse of an isomorphism is itself. Proposition 3.14(1) of [Adamek] p. 29. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
invfval.b | ⊢ 𝐵 = (Base‘𝐶) |
invfval.n | ⊢ 𝑁 = (Inv‘𝐶) |
invfval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
invfval.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
invfval.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
isoval.n | ⊢ 𝐼 = (Iso‘𝐶) |
invinv.f | ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) |
Ref | Expression |
---|---|
invinv | ⊢ (𝜑 → ((𝑌𝑁𝑋)‘((𝑋𝑁𝑌)‘𝐹)) = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | invfval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
2 | invfval.n | . . . 4 ⊢ 𝑁 = (Inv‘𝐶) | |
3 | invfval.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | invfval.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
5 | invfval.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
6 | 1, 2, 3, 4, 5 | invsym2 17021 | . . 3 ⊢ (𝜑 → ◡(𝑋𝑁𝑌) = (𝑌𝑁𝑋)) |
7 | 6 | fveq1d 6665 | . 2 ⊢ (𝜑 → (◡(𝑋𝑁𝑌)‘((𝑋𝑁𝑌)‘𝐹)) = ((𝑌𝑁𝑋)‘((𝑋𝑁𝑌)‘𝐹))) |
8 | isoval.n | . . . 4 ⊢ 𝐼 = (Iso‘𝐶) | |
9 | 1, 2, 3, 4, 5, 8 | invf1o 17027 | . . 3 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)–1-1-onto→(𝑌𝐼𝑋)) |
10 | invinv.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) | |
11 | f1ocnvfv1 7024 | . . 3 ⊢ (((𝑋𝑁𝑌):(𝑋𝐼𝑌)–1-1-onto→(𝑌𝐼𝑋) ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (◡(𝑋𝑁𝑌)‘((𝑋𝑁𝑌)‘𝐹)) = 𝐹) | |
12 | 9, 10, 11 | syl2anc 584 | . 2 ⊢ (𝜑 → (◡(𝑋𝑁𝑌)‘((𝑋𝑁𝑌)‘𝐹)) = 𝐹) |
13 | 7, 12 | eqtr3d 2855 | 1 ⊢ (𝜑 → ((𝑌𝑁𝑋)‘((𝑋𝑁𝑌)‘𝐹)) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ◡ccnv 5547 –1-1-onto→wf1o 6347 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 Catccat 16923 Invcinv 17003 Isociso 17004 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-cat 16927 df-cid 16928 df-sect 17005 df-inv 17006 df-iso 17007 |
This theorem is referenced by: (None) |
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