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Mirrors > Home > ILE Home > Th. List > oprabexd | GIF version |
Description: Existence of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
oprabexd.1 | ⊢ (𝜑 → 𝐴 ∈ V) |
oprabexd.2 | ⊢ (𝜑 → 𝐵 ∈ V) |
oprabexd.3 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑧𝜓) |
oprabexd.4 | ⊢ (𝜑 → 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)}) |
Ref | Expression |
---|---|
oprabexd | ⊢ (𝜑 → 𝐹 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oprabexd.4 | . 2 ⊢ (𝜑 → 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)}) | |
2 | oprabexd.3 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑧𝜓) | |
3 | 2 | ex 114 | . . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃*𝑧𝜓)) |
4 | moanimv 2072 | . . . . . 6 ⊢ (∃*𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃*𝑧𝜓)) | |
5 | 3, 4 | sylibr 133 | . . . . 5 ⊢ (𝜑 → ∃*𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)) |
6 | 5 | alrimivv 1847 | . . . 4 ⊢ (𝜑 → ∀𝑥∀𝑦∃*𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)) |
7 | funoprabg 5863 | . . . 4 ⊢ (∀𝑥∀𝑦∃*𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)}) | |
8 | 6, 7 | syl 14 | . . 3 ⊢ (𝜑 → Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)}) |
9 | dmoprabss 5846 | . . . 4 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)} ⊆ (𝐴 × 𝐵) | |
10 | oprabexd.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ V) | |
11 | oprabexd.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ V) | |
12 | xpexg 4648 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 × 𝐵) ∈ V) | |
13 | 10, 11, 12 | syl2anc 408 | . . . 4 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V) |
14 | ssexg 4062 | . . . 4 ⊢ ((dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)} ⊆ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ∈ V) → dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)} ∈ V) | |
15 | 9, 13, 14 | sylancr 410 | . . 3 ⊢ (𝜑 → dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)} ∈ V) |
16 | funex 5636 | . . 3 ⊢ ((Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)} ∧ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)} ∈ V) → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)} ∈ V) | |
17 | 8, 15, 16 | syl2anc 408 | . 2 ⊢ (𝜑 → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)} ∈ V) |
18 | 1, 17 | eqeltrd 2214 | 1 ⊢ (𝜑 → 𝐹 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1329 = wceq 1331 ∈ wcel 1480 ∃*wmo 1998 Vcvv 2681 ⊆ wss 3066 × cxp 4532 dom cdm 4534 Fun wfun 5112 {coprab 5768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-oprab 5771 |
This theorem is referenced by: (None) |
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