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Mirrors > Home > ILE Home > Th. List > ovigg | GIF version |
Description: The value of an operation class abstraction. Compare ovig 5900. The condition (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) is been removed. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 19-Dec-2013.) |
Ref | Expression |
---|---|
ovigg.1 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) |
ovigg.4 | ⊢ ∃*𝑧𝜑 |
ovigg.5 | ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
Ref | Expression |
---|---|
ovigg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝜓 → (𝐴𝐹𝐵) = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovigg.1 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) | |
2 | 1 | eloprabga 5866 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ↔ 𝜓)) |
3 | df-ov 5785 | . . . 4 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
4 | ovigg.5 | . . . . 5 ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | |
5 | 4 | fveq1i 5430 | . . . 4 ⊢ (𝐹‘〈𝐴, 𝐵〉) = ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}‘〈𝐴, 𝐵〉) |
6 | 3, 5 | eqtri 2161 | . . 3 ⊢ (𝐴𝐹𝐵) = ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}‘〈𝐴, 𝐵〉) |
7 | ovigg.4 | . . . . 5 ⊢ ∃*𝑧𝜑 | |
8 | 7 | funoprab 5879 | . . . 4 ⊢ Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
9 | funopfv 5469 | . . . 4 ⊢ (Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} → (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} → ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}‘〈𝐴, 𝐵〉) = 𝐶)) | |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} → ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}‘〈𝐴, 𝐵〉) = 𝐶) |
11 | 6, 10 | syl5eq 2185 | . 2 ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} → (𝐴𝐹𝐵) = 𝐶) |
12 | 2, 11 | syl6bir 163 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝜓 → (𝐴𝐹𝐵) = 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∧ w3a 963 = wceq 1332 ∈ wcel 1481 ∃*wmo 2001 〈cop 3535 Fun wfun 5125 ‘cfv 5131 (class class class)co 5782 {coprab 5783 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-ov 5785 df-oprab 5786 |
This theorem is referenced by: ovig 5900 |
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