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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 7448 | . 2 ⊢ (𝜑 → (𝑋(Homf ‘𝐶)𝑌) = (𝑋(Homf ‘𝐷)𝑌)) |
| 3 | eqid 2737 | . . 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 17735 | . 2 ⊢ (𝜑 → (𝑋(Homf ‘𝐶)𝑌) = (𝑋𝐻𝑌)) |
| 9 | eqid 2737 | . . 3 ⊢ (Homf ‘𝐷) = (Homf ‘𝐷) | |
| 10 | eqid 2737 | . . 3 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
| 11 | homfeqval.j | . . 3 ⊢ 𝐽 = (Hom ‘𝐷) | |
| 12 | 1 | homfeqbas 17739 | . . . . 5 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷)) |
| 13 | 4, 12 | eqtrid 2789 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐷)) |
| 14 | 6, 13 | eleqtrd 2843 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷)) |
| 15 | 7, 13 | eleqtrd 2843 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷)) |
| 16 | 9, 10, 11, 14, 15 | homfval 17735 | . 2 ⊢ (𝜑 → (𝑋(Homf ‘𝐷)𝑌) = (𝑋𝐽𝑌)) |
| 17 | 2, 8, 16 | 3eqtr3d 2785 | 1 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋𝐽𝑌)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Hom chom 17308 Homf chomf 17709 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-homf 17713 |
| This theorem is referenced by: comfeq 17749 comfeqval 17751 catpropd 17752 cidpropd 17753 monpropd 17781 funcpropd 17947 fullpropd 17967 natpropd 18024 xpcpropd 18253 curfpropd 18278 hofpropd 18312 oppcthinco 49088 thincpropd 49091 |
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