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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 6830 | . 2 ⊢ (𝜑 → (𝑋(Homf ‘𝐶)𝑌) = (𝑋(Homf ‘𝐷)𝑌)) |
3 | eqid 2760 | . . 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 16553 | . 2 ⊢ (𝜑 → (𝑋(Homf ‘𝐶)𝑌) = (𝑋𝐻𝑌)) |
9 | eqid 2760 | . . 3 ⊢ (Homf ‘𝐷) = (Homf ‘𝐷) | |
10 | eqid 2760 | . . 3 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
11 | homfeqval.j | . . 3 ⊢ 𝐽 = (Hom ‘𝐷) | |
12 | 1 | homfeqbas 16557 | . . . . 5 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷)) |
13 | 4, 12 | syl5eq 2806 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐷)) |
14 | 6, 13 | eleqtrd 2841 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷)) |
15 | 7, 13 | eleqtrd 2841 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷)) |
16 | 9, 10, 11, 14, 15 | homfval 16553 | . 2 ⊢ (𝜑 → (𝑋(Homf ‘𝐷)𝑌) = (𝑋𝐽𝑌)) |
17 | 2, 8, 16 | 3eqtr3d 2802 | 1 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋𝐽𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1632 ∈ wcel 2139 ‘cfv 6049 (class class class)co 6813 Basecbs 16059 Hom chom 16154 Homf chomf 16528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-1st 7333 df-2nd 7334 df-homf 16532 |
This theorem is referenced by: comfeq 16567 comfeqval 16569 catpropd 16570 cidpropd 16571 monpropd 16598 funcpropd 16761 fullpropd 16781 natpropd 16837 xpcpropd 17049 curfpropd 17074 hofpropd 17108 |
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