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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 5899 | . . 3 ⊢ (𝑥𝐹𝑦) = (𝐹‘〈𝑥, 𝑦〉) | |
2 | 1 | eqeq1i 2197 | . 2 ⊢ ((𝑥𝐹𝑦) = 𝑧 ↔ (𝐹‘〈𝑥, 𝑦〉) = 𝑧) |
3 | ovid.1 | . . . . . 6 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ∃!𝑧𝜑) | |
4 | 3 | fnoprab 5999 | . . . . 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 5329 | . . . . 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 4275 | . . . . 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 5578 | . . . 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 2256 | . . . . 5 ⊢ (〈〈𝑥, 𝑦〉, 𝑧〉 ∈ 𝐹 ↔ 〈〈𝑥, 𝑦〉, 𝑧〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}) |
13 | oprabid 5928 | . . . . 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 920 | . . 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 1364 ∃!weu 2038 ∈ wcel 2160 〈cop 3610 {copab 4078 Fn wfn 5230 ‘cfv 5235 (class class class)co 5896 {coprab 5897 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-setind 4554 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fn 5238 df-fv 5243 df-ov 5899 df-oprab 5900 |
This theorem is referenced by: (None) |
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