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Mirrors > Home > MPE Home > Th. List > eqfnov | Structured version Visualization version GIF version |
Description: Equality of two operations is determined by their values. (Contributed by NM, 1-Sep-2005.) |
Ref | Expression |
---|---|
eqfnov | ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐺 Fn (𝐶 × 𝐷)) → (𝐹 = 𝐺 ↔ ((𝐴 × 𝐵) = (𝐶 × 𝐷) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqfnfv2 6220 | . 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐺 Fn (𝐶 × 𝐷)) → (𝐹 = 𝐺 ↔ ((𝐴 × 𝐵) = (𝐶 × 𝐷) ∧ ∀𝑧 ∈ (𝐴 × 𝐵)(𝐹‘𝑧) = (𝐺‘𝑧)))) | |
2 | fveq2 6103 | . . . . . 6 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐹‘𝑧) = (𝐹‘〈𝑥, 𝑦〉)) | |
3 | fveq2 6103 | . . . . . 6 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐺‘𝑧) = (𝐺‘〈𝑥, 𝑦〉)) | |
4 | 2, 3 | eqeq12d 2625 | . . . . 5 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘〈𝑥, 𝑦〉) = (𝐺‘〈𝑥, 𝑦〉))) |
5 | df-ov 6552 | . . . . . 6 ⊢ (𝑥𝐹𝑦) = (𝐹‘〈𝑥, 𝑦〉) | |
6 | df-ov 6552 | . . . . . 6 ⊢ (𝑥𝐺𝑦) = (𝐺‘〈𝑥, 𝑦〉) | |
7 | 5, 6 | eqeq12i 2624 | . . . . 5 ⊢ ((𝑥𝐹𝑦) = (𝑥𝐺𝑦) ↔ (𝐹‘〈𝑥, 𝑦〉) = (𝐺‘〈𝑥, 𝑦〉)) |
8 | 4, 7 | syl6bbr 277 | . . . 4 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝑥𝐹𝑦) = (𝑥𝐺𝑦))) |
9 | 8 | ralxp 5185 | . . 3 ⊢ (∀𝑧 ∈ (𝐴 × 𝐵)(𝐹‘𝑧) = (𝐺‘𝑧) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦)) |
10 | 9 | anbi2i 726 | . 2 ⊢ (((𝐴 × 𝐵) = (𝐶 × 𝐷) ∧ ∀𝑧 ∈ (𝐴 × 𝐵)(𝐹‘𝑧) = (𝐺‘𝑧)) ↔ ((𝐴 × 𝐵) = (𝐶 × 𝐷) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦))) |
11 | 1, 10 | syl6bb 275 | 1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐺 Fn (𝐶 × 𝐷)) → (𝐹 = 𝐺 ↔ ((𝐴 × 𝐵) = (𝐶 × 𝐷) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∀wral 2896 〈cop 4131 × cxp 5036 Fn wfn 5799 ‘cfv 5804 (class class class)co 6549 |
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-iun 4457 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 df-ov 6552 |
This theorem is referenced by: eqfnov2 6665 oprres 6700 ssceq 16309 sspg 26967 ssps 26969 sspmlem 26971 |
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