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Mirrors > Home > MPE Home > Th. List > eqfnfv3 | Structured version Visualization version GIF version |
Description: Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
eqfnfv3 | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ (𝐹‘𝑥) = (𝐺‘𝑥))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqfnfv2 6220 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))) | |
2 | eqss 3583 | . . . . 5 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
3 | ancom 465 | . . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) ↔ (𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵)) | |
4 | 2, 3 | bitri 263 | . . . 4 ⊢ (𝐴 = 𝐵 ↔ (𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵)) |
5 | 4 | anbi1i 727 | . . 3 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)) ↔ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) |
6 | anass 679 | . . . 4 ⊢ (((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)) ↔ (𝐵 ⊆ 𝐴 ∧ (𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))) | |
7 | dfss3 3558 | . . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) | |
8 | 7 | anbi1i 727 | . . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)) ↔ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) |
9 | r19.26 3046 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ (𝐹‘𝑥) = (𝐺‘𝑥)) ↔ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) | |
10 | 8, 9 | bitr4i 266 | . . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) |
11 | 10 | anbi2i 726 | . . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ (𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ (𝐹‘𝑥) = (𝐺‘𝑥)))) |
12 | 6, 11 | bitri 263 | . . 3 ⊢ (((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)) ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ (𝐹‘𝑥) = (𝐺‘𝑥)))) |
13 | 5, 12 | bitri 263 | . 2 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)) ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ (𝐹‘𝑥) = (𝐺‘𝑥)))) |
14 | 1, 13 | syl6bb 275 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ (𝐹‘𝑥) = (𝐺‘𝑥))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⊆ wss 3540 Fn wfn 5799 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 |
This theorem is referenced by: (None) |
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