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Mirrors > Home > ILE Home > Th. List > ovelimab | GIF version |
Description: Operation value in an image. (Contributed by Mario Carneiro, 23-Dec-2013.) (Revised by Mario Carneiro, 29-Jan-2014.) |
Ref | Expression |
---|---|
ovelimab | ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (𝐷 ∈ (𝐹 “ (𝐵 × 𝐶)) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐷 = (𝑥𝐹𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvelimab 5550 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (𝐷 ∈ (𝐹 “ (𝐵 × 𝐶)) ↔ ∃𝑧 ∈ (𝐵 × 𝐶)(𝐹‘𝑧) = 𝐷)) | |
2 | fveq2 5494 | . . . . . 6 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐹‘𝑧) = (𝐹‘〈𝑥, 𝑦〉)) | |
3 | df-ov 5854 | . . . . . 6 ⊢ (𝑥𝐹𝑦) = (𝐹‘〈𝑥, 𝑦〉) | |
4 | 2, 3 | eqtr4di 2221 | . . . . 5 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐹‘𝑧) = (𝑥𝐹𝑦)) |
5 | 4 | eqeq1d 2179 | . . . 4 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → ((𝐹‘𝑧) = 𝐷 ↔ (𝑥𝐹𝑦) = 𝐷)) |
6 | eqcom 2172 | . . . 4 ⊢ ((𝑥𝐹𝑦) = 𝐷 ↔ 𝐷 = (𝑥𝐹𝑦)) | |
7 | 5, 6 | bitrdi 195 | . . 3 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → ((𝐹‘𝑧) = 𝐷 ↔ 𝐷 = (𝑥𝐹𝑦))) |
8 | 7 | rexxp 4753 | . 2 ⊢ (∃𝑧 ∈ (𝐵 × 𝐶)(𝐹‘𝑧) = 𝐷 ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐷 = (𝑥𝐹𝑦)) |
9 | 1, 8 | bitrdi 195 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (𝐷 ∈ (𝐹 “ (𝐵 × 𝐶)) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐷 = (𝑥𝐹𝑦))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1348 ∈ wcel 2141 ∃wrex 2449 ⊆ wss 3121 〈cop 3584 × cxp 4607 “ cima 4612 Fn wfn 5191 ‘cfv 5196 (class class class)co 5851 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-iun 3873 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-fv 5204 df-ov 5854 |
This theorem is referenced by: dfz2 9277 elq 9574 |
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