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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 5906 | . . . 4 ⊢ (𝜑 → dom (Homf ‘𝐶) = dom (Homf ‘𝐷)) |
3 | eqid 2733 | . . . . . 6 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
4 | eqid 2733 | . . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
5 | 3, 4 | homffn 17637 | . . . . 5 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) |
6 | 5 | fndmi 6654 | . . . 4 ⊢ dom (Homf ‘𝐶) = ((Base‘𝐶) × (Base‘𝐶)) |
7 | eqid 2733 | . . . . . 6 ⊢ (Homf ‘𝐷) = (Homf ‘𝐷) | |
8 | eqid 2733 | . . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
9 | 7, 8 | homffn 17637 | . . . . 5 ⊢ (Homf ‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷)) |
10 | 9 | fndmi 6654 | . . . 4 ⊢ dom (Homf ‘𝐷) = ((Base‘𝐷) × (Base‘𝐷)) |
11 | 2, 6, 10 | 3eqtr3g 2796 | . . 3 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ((Base‘𝐷) × (Base‘𝐷))) |
12 | 11 | dmeqd 5906 | . 2 ⊢ (𝜑 → dom ((Base‘𝐶) × (Base‘𝐶)) = dom ((Base‘𝐷) × (Base‘𝐷))) |
13 | dmxpid 5930 | . 2 ⊢ dom ((Base‘𝐶) × (Base‘𝐶)) = (Base‘𝐶) | |
14 | dmxpid 5930 | . 2 ⊢ dom ((Base‘𝐷) × (Base‘𝐷)) = (Base‘𝐷) | |
15 | 12, 13, 14 | 3eqtr3g 2796 | 1 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 × cxp 5675 dom cdm 5677 ‘cfv 6544 Basecbs 17144 Homf chomf 17610 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-homf 17614 |
This theorem is referenced by: homfeqval 17641 comfeqd 17651 comfeqval 17652 catpropd 17653 cidpropd 17654 oppccomfpropd 17673 monpropd 17684 funcpropd 17851 fullpropd 17871 fthpropd 17872 natpropd 17929 fucpropd 17930 xpcpropd 18161 curfpropd 18186 hofpropd 18220 |
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