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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 5515 | . . 3 ⊢ (𝑥𝐹𝑦) = (𝐹‘〈𝑥, 𝑦〉) | |
2 | 1 | eqeq1i 2047 | . 2 ⊢ ((𝑥𝐹𝑦) = 𝑧 ↔ (𝐹‘〈𝑥, 𝑦〉) = 𝑧) |
3 | ovid.1 | . . . . . 6 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ∃!𝑧𝜑) | |
4 | 3 | fnoprab 5604 | . . . . 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 4993 | . . . . 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 134 | . . . 4 ⊢ 𝐹 Fn {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)} |
8 | opabid 3994 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)} ↔ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)) | |
9 | 8 | biimpri 124 | . . . 4 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)}) |
10 | fnopfvb 5215 | . . . 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 393 | . . 3 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ((𝐹‘〈𝑥, 𝑦〉) = 𝑧 ↔ 〈〈𝑥, 𝑦〉, 𝑧〉 ∈ 𝐹)) |
12 | 5 | eleq2i 2104 | . . . . 5 ⊢ (〈〈𝑥, 𝑦〉, 𝑧〉 ∈ 𝐹 ↔ 〈〈𝑥, 𝑦〉, 𝑧〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}) |
13 | oprabid 5537 | . . . . 5 ⊢ (〈〈𝑥, 𝑦〉, 𝑧〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)} ↔ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)) | |
14 | 12, 13 | bitri 173 | . . . 4 ⊢ (〈〈𝑥, 𝑦〉, 𝑧〉 ∈ 𝐹 ↔ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)) |
15 | 14 | baib 828 | . . 3 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → (〈〈𝑥, 𝑦〉, 𝑧〉 ∈ 𝐹 ↔ 𝜑)) |
16 | 11, 15 | bitrd 177 | . 2 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ((𝐹‘〈𝑥, 𝑦〉) = 𝑧 ↔ 𝜑)) |
17 | 2, 16 | syl5bb 181 | 1 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ((𝑥𝐹𝑦) = 𝑧 ↔ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 ∈ wcel 1393 ∃!weu 1900 〈cop 3378 {copab 3817 Fn wfn 4897 ‘cfv 4902 (class class class)co 5512 {coprab 5513 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fn 4905 df-fv 4910 df-ov 5515 df-oprab 5516 |
This theorem is referenced by: (None) |
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