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Mirrors > Home > MPE Home > Th. List > funciso | Structured version Visualization version GIF version |
Description: The image of an isomorphism under a functor is an isomorphism. Proposition 3.21 of [Adamek] p. 32. (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
funciso.b | ⊢ 𝐵 = (Base‘𝐷) |
funciso.s | ⊢ 𝐼 = (Iso‘𝐷) |
funciso.t | ⊢ 𝐽 = (Iso‘𝐸) |
funciso.f | ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) |
funciso.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
funciso.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
funciso.m | ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐼𝑌)) |
Ref | Expression |
---|---|
funciso | ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . 2 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
2 | eqid 2733 | . 2 ⊢ (Inv‘𝐸) = (Inv‘𝐸) | |
3 | funciso.f | . . . . 5 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) | |
4 | df-br 5148 | . . . . 5 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) | |
5 | 3, 4 | sylib 217 | . . . 4 ⊢ (𝜑 → 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) |
6 | funcrcl 17809 | . . . 4 ⊢ (〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) |
8 | 7 | simprd 497 | . 2 ⊢ (𝜑 → 𝐸 ∈ Cat) |
9 | funciso.b | . . . 4 ⊢ 𝐵 = (Base‘𝐷) | |
10 | 9, 1, 3 | funcf1 17812 | . . 3 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐸)) |
11 | funciso.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
12 | 10, 11 | ffvelcdmd 7083 | . 2 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐸)) |
13 | funciso.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
14 | 10, 13 | ffvelcdmd 7083 | . 2 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐸)) |
15 | funciso.t | . 2 ⊢ 𝐽 = (Iso‘𝐸) | |
16 | eqid 2733 | . . 3 ⊢ (Inv‘𝐷) = (Inv‘𝐷) | |
17 | funciso.s | . . . 4 ⊢ 𝐼 = (Iso‘𝐷) | |
18 | 7 | simpld 496 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat) |
19 | funciso.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐼𝑌)) | |
20 | 9, 17, 16, 18, 11, 13, 19 | invisoinvr 17734 | . . 3 ⊢ (𝜑 → 𝑀(𝑋(Inv‘𝐷)𝑌)((𝑋(Inv‘𝐷)𝑌)‘𝑀)) |
21 | 9, 16, 2, 3, 11, 13, 20 | funcinv 17819 | . 2 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)(Inv‘𝐸)(𝐹‘𝑌))((𝑌𝐺𝑋)‘((𝑋(Inv‘𝐷)𝑌)‘𝑀))) |
22 | 1, 2, 8, 12, 14, 15, 21 | inviso1 17709 | 1 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 〈cop 4633 class class class wbr 5147 ‘cfv 6540 (class class class)co 7404 Basecbs 17140 Catccat 17604 Invcinv 17688 Isociso 17689 Func cfunc 17800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7970 df-2nd 7971 df-map 8818 df-ixp 8888 df-cat 17608 df-cid 17609 df-sect 17690 df-inv 17691 df-iso 17692 df-func 17804 |
This theorem is referenced by: ffthiso 17876 |
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