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| Mirrors > Home > MPE Home > Th. List > eqfnfv2f | Structured version Visualization version GIF version | ||
| Description: Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). This version of eqfnfv 7051 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.) |
| Ref | Expression |
|---|---|
| eqfnfv2f.1 | ⊢ Ⅎ𝑥𝐹 |
| eqfnfv2f.2 | ⊢ Ⅎ𝑥𝐺 |
| Ref | Expression |
|---|---|
| eqfnfv2f | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqfnfv 7051 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧))) | |
| 2 | eqfnfv2f.1 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
| 3 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑥𝑧 | |
| 4 | 2, 3 | nffv 6916 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧) |
| 5 | eqfnfv2f.2 | . . . . 5 ⊢ Ⅎ𝑥𝐺 | |
| 6 | 5, 3 | nffv 6916 | . . . 4 ⊢ Ⅎ𝑥(𝐺‘𝑧) |
| 7 | 4, 6 | nfeq 2919 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧) = (𝐺‘𝑧) |
| 8 | nfv 1914 | . . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥) = (𝐺‘𝑥) | |
| 9 | fveq2 6906 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) | |
| 10 | fveq2 6906 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝐺‘𝑧) = (𝐺‘𝑥)) | |
| 11 | 9, 10 | eqeq12d 2753 | . . 3 ⊢ (𝑧 = 𝑥 → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘𝑥) = (𝐺‘𝑥))) |
| 12 | 7, 8, 11 | cbvralw 3306 | . 2 ⊢ (∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)) |
| 13 | 1, 12 | bitrdi 287 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 Ⅎwnfc 2890 ∀wral 3061 Fn wfn 6556 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 |
| This theorem is referenced by: aacllem 49320 |
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