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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 6821. 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 6693 | . 2 ⊢ (𝑥𝐹𝑦) = (𝐹‘〈𝑥, 𝑦〉) | |
2 | ovidig.1 | . . . . 5 ⊢ ∃*𝑧𝜑 | |
3 | 2 | funoprab 6802 | . . . 4 ⊢ Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
4 | ovidig.2 | . . . . 5 ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | |
5 | 4 | funeqi 5947 | . . . 4 ⊢ (Fun 𝐹 ↔ Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}) |
6 | 3, 5 | mpbir 221 | . . 3 ⊢ Fun 𝐹 |
7 | oprabid 6717 | . . . . 5 ⊢ (〈〈𝑥, 𝑦〉, 𝑧〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ↔ 𝜑) | |
8 | 7 | biimpri 218 | . . . 4 ⊢ (𝜑 → 〈〈𝑥, 𝑦〉, 𝑧〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}) |
9 | 8, 4 | syl6eleqr 2741 | . . 3 ⊢ (𝜑 → 〈〈𝑥, 𝑦〉, 𝑧〉 ∈ 𝐹) |
10 | funopfv 6273 | . . 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 | syl5eq 2697 | 1 ⊢ (𝜑 → (𝑥𝐹𝑦) = 𝑧) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 ∃*wmo 2499 〈cop 4216 Fun wfun 5920 ‘cfv 5926 (class class class)co 6690 {coprab 6691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-iota 5889 df-fun 5928 df-fv 5934 df-ov 6693 df-oprab 6694 |
This theorem is referenced by: ovidi 6821 |
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