![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7373 | . 2 ⊢ (𝜑 → (𝑋(Homf ‘𝐶)𝑌) = (𝑋(Homf ‘𝐷)𝑌)) |
3 | eqid 2736 | . . 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 17571 | . 2 ⊢ (𝜑 → (𝑋(Homf ‘𝐶)𝑌) = (𝑋𝐻𝑌)) |
9 | eqid 2736 | . . 3 ⊢ (Homf ‘𝐷) = (Homf ‘𝐷) | |
10 | eqid 2736 | . . 3 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
11 | homfeqval.j | . . 3 ⊢ 𝐽 = (Hom ‘𝐷) | |
12 | 1 | homfeqbas 17575 | . . . . 5 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷)) |
13 | 4, 12 | eqtrid 2788 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐷)) |
14 | 6, 13 | eleqtrd 2840 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷)) |
15 | 7, 13 | eleqtrd 2840 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷)) |
16 | 9, 10, 11, 14, 15 | homfval 17571 | . 2 ⊢ (𝜑 → (𝑋(Homf ‘𝐷)𝑌) = (𝑋𝐽𝑌)) |
17 | 2, 8, 16 | 3eqtr3d 2784 | 1 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋𝐽𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ‘cfv 6496 (class class class)co 7356 Basecbs 17082 Hom chom 17143 Homf chomf 17545 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7359 df-oprab 7360 df-mpo 7361 df-1st 7920 df-2nd 7921 df-homf 17549 |
This theorem is referenced by: comfeq 17585 comfeqval 17587 catpropd 17588 cidpropd 17589 monpropd 17619 funcpropd 17786 fullpropd 17806 natpropd 17864 xpcpropd 18096 curfpropd 18121 hofpropd 18155 |
Copyright terms: Public domain | W3C validator |