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Mirrors > Home > MPE Home > Th. List > ovid | Structured version Visualization version GIF version |
Description: The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.) |
Ref | Expression |
---|---|
ovid.1 | ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ∃!𝑧𝜑) |
ovid.2 | ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)} |
Ref | Expression |
---|---|
ovid | ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ((𝑥𝐹𝑦) = 𝑧 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 7434 | . . 3 ⊢ (𝑥𝐹𝑦) = (𝐹‘〈𝑥, 𝑦〉) | |
2 | 1 | eqeq1i 2740 | . 2 ⊢ ((𝑥𝐹𝑦) = 𝑧 ↔ (𝐹‘〈𝑥, 𝑦〉) = 𝑧) |
3 | ovid.1 | . . . . . 6 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ∃!𝑧𝜑) | |
4 | 3 | fnoprab 7558 | . . . . 5 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)} Fn {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)} |
5 | ovid.2 | . . . . . 6 ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)} | |
6 | 5 | fneq1i 6666 | . . . . 5 ⊢ (𝐹 Fn {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)} ↔ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)} Fn {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)}) |
7 | 4, 6 | mpbir 231 | . . . 4 ⊢ 𝐹 Fn {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)} |
8 | opabidw 5534 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)} ↔ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)) | |
9 | 8 | biimpri 228 | . . . 4 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)}) |
10 | fnopfvb 6961 | . . . 4 ⊢ ((𝐹 Fn {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)} ∧ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)}) → ((𝐹‘〈𝑥, 𝑦〉) = 𝑧 ↔ 〈〈𝑥, 𝑦〉, 𝑧〉 ∈ 𝐹)) | |
11 | 7, 9, 10 | sylancr 587 | . . 3 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ((𝐹‘〈𝑥, 𝑦〉) = 𝑧 ↔ 〈〈𝑥, 𝑦〉, 𝑧〉 ∈ 𝐹)) |
12 | 5 | eleq2i 2831 | . . . . 5 ⊢ (〈〈𝑥, 𝑦〉, 𝑧〉 ∈ 𝐹 ↔ 〈〈𝑥, 𝑦〉, 𝑧〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}) |
13 | oprabidw 7462 | . . . . 5 ⊢ (〈〈𝑥, 𝑦〉, 𝑧〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)} ↔ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)) | |
14 | 12, 13 | bitri 275 | . . . 4 ⊢ (〈〈𝑥, 𝑦〉, 𝑧〉 ∈ 𝐹 ↔ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)) |
15 | 14 | baib 535 | . . 3 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → (〈〈𝑥, 𝑦〉, 𝑧〉 ∈ 𝐹 ↔ 𝜑)) |
16 | 11, 15 | bitrd 279 | . 2 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ((𝐹‘〈𝑥, 𝑦〉) = 𝑧 ↔ 𝜑)) |
17 | 2, 16 | bitrid 283 | 1 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ((𝑥𝐹𝑦) = 𝑧 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃!weu 2566 〈cop 4637 {copab 5210 Fn wfn 6558 ‘cfv 6563 (class class class)co 7431 {coprab 7432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fn 6566 df-fv 6571 df-ov 7434 df-oprab 7435 |
This theorem is referenced by: (None) |
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