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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 5774 | . . . 4 ⊢ (𝜑 → dom (Homf ‘𝐶) = dom (Homf ‘𝐷)) |
3 | eqid 2737 | . . . . . 6 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
4 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
5 | 3, 4 | homffn 17196 | . . . . 5 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) |
6 | 5 | fndmi 6482 | . . . 4 ⊢ dom (Homf ‘𝐶) = ((Base‘𝐶) × (Base‘𝐶)) |
7 | eqid 2737 | . . . . . 6 ⊢ (Homf ‘𝐷) = (Homf ‘𝐷) | |
8 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
9 | 7, 8 | homffn 17196 | . . . . 5 ⊢ (Homf ‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷)) |
10 | 9 | fndmi 6482 | . . . 4 ⊢ dom (Homf ‘𝐷) = ((Base‘𝐷) × (Base‘𝐷)) |
11 | 2, 6, 10 | 3eqtr3g 2801 | . . 3 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ((Base‘𝐷) × (Base‘𝐷))) |
12 | 11 | dmeqd 5774 | . 2 ⊢ (𝜑 → dom ((Base‘𝐶) × (Base‘𝐶)) = dom ((Base‘𝐷) × (Base‘𝐷))) |
13 | dmxpid 5799 | . 2 ⊢ dom ((Base‘𝐶) × (Base‘𝐶)) = (Base‘𝐶) | |
14 | dmxpid 5799 | . 2 ⊢ dom ((Base‘𝐷) × (Base‘𝐷)) = (Base‘𝐷) | |
15 | 12, 13, 14 | 3eqtr3g 2801 | 1 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 × cxp 5549 dom cdm 5551 ‘cfv 6380 Basecbs 16760 Homf chomf 17169 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-1st 7761 df-2nd 7762 df-homf 17173 |
This theorem is referenced by: homfeqval 17200 comfeqd 17210 comfeqval 17211 catpropd 17212 cidpropd 17213 oppccomfpropd 17231 monpropd 17242 funcpropd 17407 fullpropd 17427 fthpropd 17428 natpropd 17485 fucpropd 17486 xpcpropd 17716 curfpropd 17741 hofpropd 17775 |
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