![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > ovigg | GIF version |
Description: The value of an operation class abstraction. Compare ovig 5993. The condition (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) is been removed. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 19-Dec-2013.) |
Ref | Expression |
---|---|
ovigg.1 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) |
ovigg.4 | ⊢ ∃*𝑧𝜑 |
ovigg.5 | ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
Ref | Expression |
---|---|
ovigg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝜓 → (𝐴𝐹𝐵) = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovigg.1 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) | |
2 | 1 | eloprabga 5959 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ↔ 𝜓)) |
3 | df-ov 5875 | . . . 4 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
4 | ovigg.5 | . . . . 5 ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | |
5 | 4 | fveq1i 5515 | . . . 4 ⊢ (𝐹‘〈𝐴, 𝐵〉) = ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}‘〈𝐴, 𝐵〉) |
6 | 3, 5 | eqtri 2198 | . . 3 ⊢ (𝐴𝐹𝐵) = ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}‘〈𝐴, 𝐵〉) |
7 | ovigg.4 | . . . . 5 ⊢ ∃*𝑧𝜑 | |
8 | 7 | funoprab 5972 | . . . 4 ⊢ Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
9 | funopfv 5554 | . . . 4 ⊢ (Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} → (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} → ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}‘〈𝐴, 𝐵〉) = 𝐶)) | |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} → ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}‘〈𝐴, 𝐵〉) = 𝐶) |
11 | 6, 10 | eqtrid 2222 | . 2 ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} → (𝐴𝐹𝐵) = 𝐶) |
12 | 2, 11 | syl6bir 164 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝜓 → (𝐴𝐹𝐵) = 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∧ w3a 978 = wceq 1353 ∃*wmo 2027 ∈ wcel 2148 〈cop 3595 Fun wfun 5209 ‘cfv 5215 (class class class)co 5872 {coprab 5873 |
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-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 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 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-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 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 5177 df-fun 5217 df-fv 5223 df-ov 5875 df-oprab 5876 |
This theorem is referenced by: ovig 5993 |
Copyright terms: Public domain | W3C validator |