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Mirrors > Home > MPE Home > Th. List > ovigg | Structured version Visualization version GIF version |
Description: The value of an operation class abstraction. Compare ovig 6680. 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 6645 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ↔ 𝜓)) |
3 | df-ov 6552 | . . . 4 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
4 | ovigg.5 | . . . . 5 ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | |
5 | 4 | fveq1i 6104 | . . . 4 ⊢ (𝐹‘〈𝐴, 𝐵〉) = ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}‘〈𝐴, 𝐵〉) |
6 | 3, 5 | eqtri 2632 | . . 3 ⊢ (𝐴𝐹𝐵) = ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}‘〈𝐴, 𝐵〉) |
7 | ovigg.4 | . . . . 5 ⊢ ∃*𝑧𝜑 | |
8 | 7 | funoprab 6658 | . . . 4 ⊢ Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
9 | funopfv 6145 | . . . 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 2656 | . 2 ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} → (𝐴𝐹𝐵) = 𝐶) |
12 | 2, 11 | syl6bir 243 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝜓 → (𝐴𝐹𝐵) = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∃*wmo 2459 〈cop 4131 Fun wfun 5798 ‘cfv 5804 (class class class)co 6549 {coprab 6550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 |
This theorem is referenced by: ovig 6680 joinval 16828 meetval 16842 |
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