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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqfnfv2d2 | Structured version Visualization version GIF version |
Description: Equality of functions is determined by their values, a deduction version. (Contributed by metakunt, 28-May-2024.) |
Ref | Expression |
---|---|
eqfnfv2d2.1 | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
eqfnfv2d2.2 | ⊢ (𝜑 → 𝐺 Fn 𝐵) |
eqfnfv2d2.3 | ⊢ (𝜑 → 𝐴 = 𝐵) |
eqfnfv2d2.4 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥)) |
Ref | Expression |
---|---|
eqfnfv2d2 | ⊢ (𝜑 → 𝐹 = 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqfnfv2d2.3 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | eqfnfv2d2.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥)) | |
3 | 2 | ralrimiva 3145 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)) |
4 | 1, 3 | jca 511 | . 2 ⊢ (𝜑 → (𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) |
5 | eqfnfv2d2.1 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
6 | eqfnfv2d2.2 | . . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐵) | |
7 | 5, 6 | jca 511 | . . 3 ⊢ (𝜑 → (𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵)) |
8 | eqfnfv2 7050 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (𝜑 → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))) |
10 | 4, 9 | mpbird 257 | 1 ⊢ (𝜑 → 𝐹 = 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3060 Fn wfn 6554 ‘cfv 6559 |
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 2707 ax-sep 5294 ax-nul 5304 ax-pr 5430 |
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 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-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-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-opab 5204 df-mpt 5224 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-iota 6512 df-fun 6561 df-fn 6562 df-fv 6567 |
This theorem is referenced by: metakunt25 42208 |
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