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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 3144 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)) |
4 | 1, 3 | jca 513 | . 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 513 | . . 3 ⊢ (𝜑 → (𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵)) |
8 | eqfnfv2 6988 | . . 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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3065 Fn wfn 6496 ‘cfv 6501 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-fv 6509 |
This theorem is referenced by: metakunt25 40630 |
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