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| 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 5915 | . . . 4 ⊢ (𝜑 → dom (Homf ‘𝐶) = dom (Homf ‘𝐷)) | 
| 3 | eqid 2736 | . . . . . 6 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
| 4 | eqid 2736 | . . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
| 5 | 3, 4 | homffn 17737 | . . . . 5 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) | 
| 6 | 5 | fndmi 6671 | . . . 4 ⊢ dom (Homf ‘𝐶) = ((Base‘𝐶) × (Base‘𝐶)) | 
| 7 | eqid 2736 | . . . . . 6 ⊢ (Homf ‘𝐷) = (Homf ‘𝐷) | |
| 8 | eqid 2736 | . . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
| 9 | 7, 8 | homffn 17737 | . . . . 5 ⊢ (Homf ‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷)) | 
| 10 | 9 | fndmi 6671 | . . . 4 ⊢ dom (Homf ‘𝐷) = ((Base‘𝐷) × (Base‘𝐷)) | 
| 11 | 2, 6, 10 | 3eqtr3g 2799 | . . 3 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ((Base‘𝐷) × (Base‘𝐷))) | 
| 12 | 11 | dmeqd 5915 | . 2 ⊢ (𝜑 → dom ((Base‘𝐶) × (Base‘𝐶)) = dom ((Base‘𝐷) × (Base‘𝐷))) | 
| 13 | dmxpid 5940 | . 2 ⊢ dom ((Base‘𝐶) × (Base‘𝐶)) = (Base‘𝐶) | |
| 14 | dmxpid 5940 | . 2 ⊢ dom ((Base‘𝐷) × (Base‘𝐷)) = (Base‘𝐷) | |
| 15 | 12, 13, 14 | 3eqtr3g 2799 | 1 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 × cxp 5682 dom cdm 5684 ‘cfv 6560 Basecbs 17248 Homf chomf 17710 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-homf 17714 | 
| This theorem is referenced by: homfeqval 17741 comfeqd 17751 comfeqval 17752 catpropd 17753 cidpropd 17754 oppccomfpropd 17771 monpropd 17782 funcpropd 17948 fullpropd 17968 fthpropd 17969 natpropd 18025 fucpropd 18026 xpcpropd 18254 curfpropd 18279 hofpropd 18313 oppcthinco 49113 thincpropd 49116 termcpropd 49163 | 
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