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Mirrors > Home > MPE Home > Th. List > homfeqbas | Structured version Visualization version GIF version |
Description: Deduce equality of base sets from equality of Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
homfeqbas.1 | ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) |
Ref | Expression |
---|---|
homfeqbas | ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | homfeqbas.1 | . . . . 5 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) | |
2 | 1 | dmeqd 5911 | . . . 4 ⊢ (𝜑 → dom (Homf ‘𝐶) = dom (Homf ‘𝐷)) |
3 | eqid 2725 | . . . . . 6 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
4 | eqid 2725 | . . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
5 | 3, 4 | homffn 17701 | . . . . 5 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) |
6 | 5 | fndmi 6663 | . . . 4 ⊢ dom (Homf ‘𝐶) = ((Base‘𝐶) × (Base‘𝐶)) |
7 | eqid 2725 | . . . . . 6 ⊢ (Homf ‘𝐷) = (Homf ‘𝐷) | |
8 | eqid 2725 | . . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
9 | 7, 8 | homffn 17701 | . . . . 5 ⊢ (Homf ‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷)) |
10 | 9 | fndmi 6663 | . . . 4 ⊢ dom (Homf ‘𝐷) = ((Base‘𝐷) × (Base‘𝐷)) |
11 | 2, 6, 10 | 3eqtr3g 2788 | . . 3 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ((Base‘𝐷) × (Base‘𝐷))) |
12 | 11 | dmeqd 5911 | . 2 ⊢ (𝜑 → dom ((Base‘𝐶) × (Base‘𝐶)) = dom ((Base‘𝐷) × (Base‘𝐷))) |
13 | dmxpid 5935 | . 2 ⊢ dom ((Base‘𝐶) × (Base‘𝐶)) = (Base‘𝐶) | |
14 | dmxpid 5935 | . 2 ⊢ dom ((Base‘𝐷) × (Base‘𝐷)) = (Base‘𝐷) | |
15 | 12, 13, 14 | 3eqtr3g 2788 | 1 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 × cxp 5679 dom cdm 5681 ‘cfv 6553 Basecbs 17208 Homf chomf 17674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5579 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7426 df-oprab 7427 df-mpo 7428 df-1st 8002 df-2nd 8003 df-homf 17678 |
This theorem is referenced by: homfeqval 17705 comfeqd 17715 comfeqval 17716 catpropd 17717 cidpropd 17718 oppccomfpropd 17737 monpropd 17748 funcpropd 17917 fullpropd 17937 fthpropd 17938 natpropd 17996 fucpropd 17997 xpcpropd 18228 curfpropd 18253 hofpropd 18287 |
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