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Mirrors > Home > MPE Home > Th. List > homfeqval | Structured version Visualization version GIF version |
Description: Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
homfeqval.b | ⊢ 𝐵 = (Base‘𝐶) |
homfeqval.h | ⊢ 𝐻 = (Hom ‘𝐶) |
homfeqval.j | ⊢ 𝐽 = (Hom ‘𝐷) |
homfeqval.1 | ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) |
homfeqval.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
homfeqval.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
homfeqval | ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋𝐽𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | homfeqval.1 | . . 3 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) | |
2 | 1 | oveqd 7181 | . 2 ⊢ (𝜑 → (𝑋(Homf ‘𝐶)𝑌) = (𝑋(Homf ‘𝐷)𝑌)) |
3 | eqid 2738 | . . 3 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
4 | homfeqval.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
5 | homfeqval.h | . . 3 ⊢ 𝐻 = (Hom ‘𝐶) | |
6 | homfeqval.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
7 | homfeqval.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
8 | 3, 4, 5, 6, 7 | homfval 17059 | . 2 ⊢ (𝜑 → (𝑋(Homf ‘𝐶)𝑌) = (𝑋𝐻𝑌)) |
9 | eqid 2738 | . . 3 ⊢ (Homf ‘𝐷) = (Homf ‘𝐷) | |
10 | eqid 2738 | . . 3 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
11 | homfeqval.j | . . 3 ⊢ 𝐽 = (Hom ‘𝐷) | |
12 | 1 | homfeqbas 17063 | . . . . 5 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷)) |
13 | 4, 12 | syl5eq 2785 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐷)) |
14 | 6, 13 | eleqtrd 2835 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷)) |
15 | 7, 13 | eleqtrd 2835 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷)) |
16 | 9, 10, 11, 14, 15 | homfval 17059 | . 2 ⊢ (𝜑 → (𝑋(Homf ‘𝐷)𝑌) = (𝑋𝐽𝑌)) |
17 | 2, 8, 16 | 3eqtr3d 2781 | 1 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋𝐽𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 ‘cfv 6333 (class class class)co 7164 Basecbs 16579 Hom chom 16672 Homf chomf 17033 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-ov 7167 df-oprab 7168 df-mpo 7169 df-1st 7707 df-2nd 7708 df-homf 17037 |
This theorem is referenced by: comfeq 17073 comfeqval 17075 catpropd 17076 cidpropd 17077 monpropd 17105 funcpropd 17268 fullpropd 17288 natpropd 17344 xpcpropd 17567 curfpropd 17592 hofpropd 17626 |
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