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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 3111 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)) |
4 | 1, 3 | jca 516 | . 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 516 | . . 3 ⊢ (𝜑 → (𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵)) |
8 | eqfnfv2 6787 | . . 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 260 | 1 ⊢ (𝜑 → 𝐹 = 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1539 ∈ wcel 2112 ∀wral 3068 Fn wfn 6323 ‘cfv 6328 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pr 5291 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-ral 3073 df-rex 3074 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-br 5026 df-opab 5088 df-mpt 5106 df-id 5423 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-iota 6287 df-fun 6330 df-fn 6331 df-fv 6336 |
This theorem is referenced by: metakunt25 39656 |
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