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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 5905 | . . . 4 ⊢ (𝜑 → dom (Homf ‘𝐶) = dom (Homf ‘𝐷)) |
3 | eqid 2732 | . . . . . 6 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
4 | eqid 2732 | . . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
5 | 3, 4 | homffn 17639 | . . . . 5 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) |
6 | 5 | fndmi 6653 | . . . 4 ⊢ dom (Homf ‘𝐶) = ((Base‘𝐶) × (Base‘𝐶)) |
7 | eqid 2732 | . . . . . 6 ⊢ (Homf ‘𝐷) = (Homf ‘𝐷) | |
8 | eqid 2732 | . . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
9 | 7, 8 | homffn 17639 | . . . . 5 ⊢ (Homf ‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷)) |
10 | 9 | fndmi 6653 | . . . 4 ⊢ dom (Homf ‘𝐷) = ((Base‘𝐷) × (Base‘𝐷)) |
11 | 2, 6, 10 | 3eqtr3g 2795 | . . 3 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ((Base‘𝐷) × (Base‘𝐷))) |
12 | 11 | dmeqd 5905 | . 2 ⊢ (𝜑 → dom ((Base‘𝐶) × (Base‘𝐶)) = dom ((Base‘𝐷) × (Base‘𝐷))) |
13 | dmxpid 5929 | . 2 ⊢ dom ((Base‘𝐶) × (Base‘𝐶)) = (Base‘𝐶) | |
14 | dmxpid 5929 | . 2 ⊢ dom ((Base‘𝐷) × (Base‘𝐷)) = (Base‘𝐷) | |
15 | 12, 13, 14 | 3eqtr3g 2795 | 1 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 × cxp 5674 dom cdm 5676 ‘cfv 6543 Basecbs 17146 Homf chomf 17612 |
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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-homf 17616 |
This theorem is referenced by: homfeqval 17643 comfeqd 17653 comfeqval 17654 catpropd 17655 cidpropd 17656 oppccomfpropd 17675 monpropd 17686 funcpropd 17853 fullpropd 17873 fthpropd 17874 natpropd 17931 fucpropd 17932 xpcpropd 18163 curfpropd 18188 hofpropd 18222 |
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