![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > ovidig | GIF version |
Description: The value of an operation class abstraction. Compare ovidi 5650. 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 5546 | . 2 ⊢ (𝑥𝐹𝑦) = (𝐹‘〈𝑥, 𝑦〉) | |
2 | ovidig.1 | . . . . 5 ⊢ ∃*𝑧𝜑 | |
3 | 2 | funoprab 5632 | . . . 4 ⊢ Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
4 | ovidig.2 | . . . . 5 ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | |
5 | 4 | funeqi 4952 | . . . 4 ⊢ (Fun 𝐹 ↔ Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}) |
6 | 3, 5 | mpbir 144 | . . 3 ⊢ Fun 𝐹 |
7 | oprabid 5568 | . . . . 5 ⊢ (〈〈𝑥, 𝑦〉, 𝑧〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ↔ 𝜑) | |
8 | 7 | biimpri 131 | . . . 4 ⊢ (𝜑 → 〈〈𝑥, 𝑦〉, 𝑧〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}) |
9 | 8, 4 | syl6eleqr 2173 | . . 3 ⊢ (𝜑 → 〈〈𝑥, 𝑦〉, 𝑧〉 ∈ 𝐹) |
10 | funopfv 5245 | . . 3 ⊢ (Fun 𝐹 → (〈〈𝑥, 𝑦〉, 𝑧〉 ∈ 𝐹 → (𝐹‘〈𝑥, 𝑦〉) = 𝑧)) | |
11 | 6, 9, 10 | mpsyl 64 | . 2 ⊢ (𝜑 → (𝐹‘〈𝑥, 𝑦〉) = 𝑧) |
12 | 1, 11 | syl5eq 2126 | 1 ⊢ (𝜑 → (𝑥𝐹𝑦) = 𝑧) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 ∈ wcel 1434 ∃*wmo 1943 〈cop 3409 Fun wfun 4926 ‘cfv 4932 (class class class)co 5543 {coprab 5544 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-sep 3904 ax-pow 3956 ax-pr 3972 ax-setind 4288 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ne 2247 df-ral 2354 df-rex 2355 df-v 2604 df-sbc 2817 df-dif 2976 df-un 2978 df-in 2980 df-ss 2987 df-pw 3392 df-sn 3412 df-pr 3413 df-op 3415 df-uni 3610 df-br 3794 df-opab 3848 df-id 4056 df-xp 4377 df-rel 4378 df-cnv 4379 df-co 4380 df-dm 4381 df-iota 4897 df-fun 4934 df-fv 4940 df-ov 5546 df-oprab 5547 |
This theorem is referenced by: ovidi 5650 |
Copyright terms: Public domain | W3C validator |