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Mirrors > Home > ILE Home > Th. List > fnovim | GIF version |
Description: Representation of a function in terms of its values. (Contributed by Jim Kingdon, 16-Jan-2019.) |
Ref | Expression |
---|---|
fnovim | ⊢ (𝐹 Fn (𝐴 × 𝐵) → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffn5im 5240 | . 2 ⊢ (𝐹 Fn (𝐴 × 𝐵) → 𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹‘𝑧))) | |
2 | fveq2 5198 | . . . . 5 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐹‘𝑧) = (𝐹‘〈𝑥, 𝑦〉)) | |
3 | df-ov 5535 | . . . . 5 ⊢ (𝑥𝐹𝑦) = (𝐹‘〈𝑥, 𝑦〉) | |
4 | 2, 3 | syl6eqr 2131 | . . . 4 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐹‘𝑧) = (𝑥𝐹𝑦)) |
5 | 4 | mpt2mpt 5616 | . . 3 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹‘𝑧)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)) |
6 | 5 | eqeq2i 2091 | . 2 ⊢ (𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹‘𝑧)) ↔ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦))) |
7 | 1, 6 | sylib 120 | 1 ⊢ (𝐹 Fn (𝐴 × 𝐵) → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 〈cop 3401 ↦ cmpt 3839 × cxp 4361 Fn wfn 4917 ‘cfv 4922 (class class class)co 5532 ↦ cmpt2 5534 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-csb 2909 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fn 4925 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 |
This theorem is referenced by: dfioo2 8997 |
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