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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 5620 | . . . 4 ⊢ (𝜑 → dom (Homf ‘𝐶) = dom (Homf ‘𝐷)) |
3 | eqid 2772 | . . . . . 6 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
4 | eqid 2772 | . . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
5 | 3, 4 | homffn 16833 | . . . . 5 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) |
6 | fndm 6285 | . . . . 5 ⊢ ((Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) → dom (Homf ‘𝐶) = ((Base‘𝐶) × (Base‘𝐶))) | |
7 | 5, 6 | ax-mp 5 | . . . 4 ⊢ dom (Homf ‘𝐶) = ((Base‘𝐶) × (Base‘𝐶)) |
8 | eqid 2772 | . . . . . 6 ⊢ (Homf ‘𝐷) = (Homf ‘𝐷) | |
9 | eqid 2772 | . . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
10 | 8, 9 | homffn 16833 | . . . . 5 ⊢ (Homf ‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷)) |
11 | fndm 6285 | . . . . 5 ⊢ ((Homf ‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷)) → dom (Homf ‘𝐷) = ((Base‘𝐷) × (Base‘𝐷))) | |
12 | 10, 11 | ax-mp 5 | . . . 4 ⊢ dom (Homf ‘𝐷) = ((Base‘𝐷) × (Base‘𝐷)) |
13 | 2, 7, 12 | 3eqtr3g 2831 | . . 3 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ((Base‘𝐷) × (Base‘𝐷))) |
14 | 13 | dmeqd 5620 | . 2 ⊢ (𝜑 → dom ((Base‘𝐶) × (Base‘𝐶)) = dom ((Base‘𝐷) × (Base‘𝐷))) |
15 | dmxpid 5640 | . 2 ⊢ dom ((Base‘𝐶) × (Base‘𝐶)) = (Base‘𝐶) | |
16 | dmxpid 5640 | . 2 ⊢ dom ((Base‘𝐷) × (Base‘𝐷)) = (Base‘𝐷) | |
17 | 14, 15, 16 | 3eqtr3g 2831 | 1 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 × cxp 5401 dom cdm 5403 Fn wfn 6180 ‘cfv 6185 Basecbs 16337 Homf chomf 16807 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-ov 6977 df-oprab 6978 df-mpo 6979 df-1st 7499 df-2nd 7500 df-homf 16811 |
This theorem is referenced by: homfeqval 16837 comfeqd 16847 comfeqval 16848 catpropd 16849 cidpropd 16850 oppccomfpropd 16867 monpropd 16877 funcpropd 17040 fullpropd 17060 fthpropd 17061 natpropd 17116 fucpropd 17117 xpcpropd 17328 curfpropd 17353 hofpropd 17387 |
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