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Mirrors > Home > MPE Home > Th. List > eqfnfv2 | Structured version Visualization version GIF version |
Description: Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28. (Contributed by NM, 3-Aug-1994.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
eqfnfv2 | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmeq 5356 | . . . 4 ⊢ (𝐹 = 𝐺 → dom 𝐹 = dom 𝐺) | |
2 | fndm 6028 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
3 | fndm 6028 | . . . . 5 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵) | |
4 | 2, 3 | eqeqan12d 2667 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (dom 𝐹 = dom 𝐺 ↔ 𝐴 = 𝐵)) |
5 | 1, 4 | syl5ib 234 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 = 𝐺 → 𝐴 = 𝐵)) |
6 | 5 | pm4.71rd 668 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ 𝐹 = 𝐺))) |
7 | fneq2 6018 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐺 Fn 𝐴 ↔ 𝐺 Fn 𝐵)) | |
8 | 7 | biimparc 503 | . . . . 5 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐴 = 𝐵) → 𝐺 Fn 𝐴) |
9 | eqfnfv 6351 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) | |
10 | 8, 9 | sylan2 490 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐺 Fn 𝐵 ∧ 𝐴 = 𝐵)) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) |
11 | 10 | anassrs 681 | . . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ 𝐴 = 𝐵) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) |
12 | 11 | pm5.32da 674 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → ((𝐴 = 𝐵 ∧ 𝐹 = 𝐺) ↔ (𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))) |
13 | 6, 12 | bitrd 268 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ∀wral 2941 dom cdm 5143 Fn wfn 5921 ‘cfv 5926 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-fv 5934 |
This theorem is referenced by: eqfnfv3 6353 eqfunfv 6356 eqfnov 6808 wfr3g 7458 2ffzeq 12499 eqwrd 13379 soseq 31879 frr3g 31904 sdclem2 33668 2ffzoeq 41663 |
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