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Mirrors > Home > ILE Home > Th. List > eqfnov | 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 5635 | . 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐺 Fn (𝐶 × 𝐷)) → (𝐹 = 𝐺 ↔ ((𝐴 × 𝐵) = (𝐶 × 𝐷) ∧ ∀𝑧 ∈ (𝐴 × 𝐵)(𝐹‘𝑧) = (𝐺‘𝑧)))) | |
2 | fveq2 5534 | . . . . . 6 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐹‘𝑧) = (𝐹‘〈𝑥, 𝑦〉)) | |
3 | fveq2 5534 | . . . . . 6 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐺‘𝑧) = (𝐺‘〈𝑥, 𝑦〉)) | |
4 | 2, 3 | eqeq12d 2204 | . . . . 5 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘〈𝑥, 𝑦〉) = (𝐺‘〈𝑥, 𝑦〉))) |
5 | df-ov 5899 | . . . . . 6 ⊢ (𝑥𝐹𝑦) = (𝐹‘〈𝑥, 𝑦〉) | |
6 | df-ov 5899 | . . . . . 6 ⊢ (𝑥𝐺𝑦) = (𝐺‘〈𝑥, 𝑦〉) | |
7 | 5, 6 | eqeq12i 2203 | . . . . 5 ⊢ ((𝑥𝐹𝑦) = (𝑥𝐺𝑦) ↔ (𝐹‘〈𝑥, 𝑦〉) = (𝐺‘〈𝑥, 𝑦〉)) |
8 | 4, 7 | bitr4di 198 | . . . 4 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝑥𝐹𝑦) = (𝑥𝐺𝑦))) |
9 | 8 | ralxp 4788 | . . 3 ⊢ (∀𝑧 ∈ (𝐴 × 𝐵)(𝐹‘𝑧) = (𝐺‘𝑧) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦)) |
10 | 9 | anbi2i 457 | . 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 | bitrdi 196 | 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 set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∀wral 2468 〈cop 3610 × cxp 4642 Fn wfn 5230 ‘cfv 5235 (class class class)co 5896 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-csb 3073 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fn 5238 df-fv 5243 df-ov 5899 |
This theorem is referenced by: eqfnov2 6004 |
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