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Mirrors > Home > ILE Home > Th. List > ovidig | GIF version |
Description: The value of an operation class abstraction. Compare ovidi 5889. The condition (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) is been removed. (Contributed by Mario Carneiro, 29-Dec-2014.) |
Ref | Expression |
---|---|
ovidig.1 | ⊢ ∃*𝑧𝜑 |
ovidig.2 | ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
Ref | Expression |
---|---|
ovidig | ⊢ (𝜑 → (𝑥𝐹𝑦) = 𝑧) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 5777 | . 2 ⊢ (𝑥𝐹𝑦) = (𝐹‘〈𝑥, 𝑦〉) | |
2 | ovidig.1 | . . . . 5 ⊢ ∃*𝑧𝜑 | |
3 | 2 | funoprab 5871 | . . . 4 ⊢ Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
4 | ovidig.2 | . . . . 5 ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | |
5 | 4 | funeqi 5144 | . . . 4 ⊢ (Fun 𝐹 ↔ Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}) |
6 | 3, 5 | mpbir 145 | . . 3 ⊢ Fun 𝐹 |
7 | oprabid 5803 | . . . . 5 ⊢ (〈〈𝑥, 𝑦〉, 𝑧〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ↔ 𝜑) | |
8 | 7 | biimpri 132 | . . . 4 ⊢ (𝜑 → 〈〈𝑥, 𝑦〉, 𝑧〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}) |
9 | 8, 4 | eleqtrrdi 2233 | . . 3 ⊢ (𝜑 → 〈〈𝑥, 𝑦〉, 𝑧〉 ∈ 𝐹) |
10 | funopfv 5461 | . . 3 ⊢ (Fun 𝐹 → (〈〈𝑥, 𝑦〉, 𝑧〉 ∈ 𝐹 → (𝐹‘〈𝑥, 𝑦〉) = 𝑧)) | |
11 | 6, 9, 10 | mpsyl 65 | . 2 ⊢ (𝜑 → (𝐹‘〈𝑥, 𝑦〉) = 𝑧) |
12 | 1, 11 | syl5eq 2184 | 1 ⊢ (𝜑 → (𝑥𝐹𝑦) = 𝑧) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 ∃*wmo 2000 〈cop 3530 Fun wfun 5117 ‘cfv 5123 (class class class)co 5774 {coprab 5775 |
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-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-ov 5777 df-oprab 5778 |
This theorem is referenced by: ovidi 5889 |
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