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Mirrors > Home > ILE Home > Th. List > ovid | 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 5874 | . . 3 ⊢ (𝑥𝐹𝑦) = (𝐹‘〈𝑥, 𝑦〉) | |
2 | 1 | eqeq1i 2185 | . 2 ⊢ ((𝑥𝐹𝑦) = 𝑧 ↔ (𝐹‘〈𝑥, 𝑦〉) = 𝑧) |
3 | ovid.1 | . . . . . 6 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ∃!𝑧𝜑) | |
4 | 3 | fnoprab 5974 | . . . . 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 5308 | . . . . 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 146 | . . . 4 ⊢ 𝐹 Fn {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)} |
8 | opabid 4256 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)} ↔ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)) | |
9 | 8 | biimpri 133 | . . . 4 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)}) |
10 | fnopfvb 5555 | . . . 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 414 | . . 3 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ((𝐹‘〈𝑥, 𝑦〉) = 𝑧 ↔ 〈〈𝑥, 𝑦〉, 𝑧〉 ∈ 𝐹)) |
12 | 5 | eleq2i 2244 | . . . . 5 ⊢ (〈〈𝑥, 𝑦〉, 𝑧〉 ∈ 𝐹 ↔ 〈〈𝑥, 𝑦〉, 𝑧〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}) |
13 | oprabid 5903 | . . . . 5 ⊢ (〈〈𝑥, 𝑦〉, 𝑧〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)} ↔ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)) | |
14 | 12, 13 | bitri 184 | . . . 4 ⊢ (〈〈𝑥, 𝑦〉, 𝑧〉 ∈ 𝐹 ↔ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)) |
15 | 14 | baib 919 | . . 3 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → (〈〈𝑥, 𝑦〉, 𝑧〉 ∈ 𝐹 ↔ 𝜑)) |
16 | 11, 15 | bitrd 188 | . 2 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ((𝐹‘〈𝑥, 𝑦〉) = 𝑧 ↔ 𝜑)) |
17 | 2, 16 | bitrid 192 | 1 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ((𝑥𝐹𝑦) = 𝑧 ↔ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∃!weu 2026 ∈ wcel 2148 〈cop 3595 {copab 4062 Fn wfn 5209 ‘cfv 5214 (class class class)co 5871 {coprab 5872 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-setind 4535 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4003 df-opab 4064 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-iota 5176 df-fun 5216 df-fn 5217 df-fv 5222 df-ov 5874 df-oprab 5875 |
This theorem is referenced by: (None) |
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