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Mirrors > Home > MPE Home > Th. List > ovidig | Structured version Visualization version GIF version |
Description: The value of an operation class abstraction. Compare ovidi 7551. 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 7412 | . 2 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩) | |
2 | ovidig.1 | . . . . 5 ⊢ ∃*𝑧𝜑 | |
3 | 2 | funoprab 7530 | . . . 4 ⊢ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} |
4 | ovidig.2 | . . . . 5 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} | |
5 | 4 | funeqi 6570 | . . . 4 ⊢ (Fun 𝐹 ↔ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}) |
6 | 3, 5 | mpbir 230 | . . 3 ⊢ Fun 𝐹 |
7 | oprabidw 7440 | . . . . 5 ⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜑) | |
8 | 7 | biimpri 227 | . . . 4 ⊢ (𝜑 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}) |
9 | 8, 4 | eleqtrrdi 2845 | . . 3 ⊢ (𝜑 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹) |
10 | funopfv 6944 | . . 3 ⊢ (Fun 𝐹 → (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹 → (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)) | |
11 | 6, 9, 10 | mpsyl 68 | . 2 ⊢ (𝜑 → (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) |
12 | 1, 11 | eqtrid 2785 | 1 ⊢ (𝜑 → (𝑥𝐹𝑦) = 𝑧) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∃*wmo 2533 ⟨cop 4635 Fun wfun 6538 ‘cfv 6544 (class class class)co 7409 {coprab 7410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-ov 7412 df-oprab 7413 |
This theorem is referenced by: ovidi 7551 |
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