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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 108 | . . . . . 6 ⊢ (φ → ((x ∈ A ∧ y ∈ B) → ∃*zψ)) |
4 | moanimv 1972 | . . . . . 6 ⊢ (∃*z((x ∈ A ∧ y ∈ B) ∧ ψ) ↔ ((x ∈ A ∧ y ∈ B) → ∃*zψ)) | |
5 | 3, 4 | sylibr 137 | . . . . 5 ⊢ (φ → ∃*z((x ∈ A ∧ y ∈ B) ∧ ψ)) |
6 | 5 | alrimivv 1752 | . . . 4 ⊢ (φ → ∀x∀y∃*z((x ∈ A ∧ y ∈ B) ∧ ψ)) |
7 | funoprabg 5542 | . . . 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 5528 | . . . 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 4395 | . . . . 5 ⊢ ((A ∈ V ∧ B ∈ V) → (A × B) ∈ V) | |
13 | 10, 11, 12 | syl2anc 391 | . . . 4 ⊢ (φ → (A × B) ∈ V) |
14 | ssexg 3887 | . . . 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 393 | . . 3 ⊢ (φ → dom {〈〈x, y〉, z〉 ∣ ((x ∈ A ∧ y ∈ B) ∧ ψ)} ∈ V) |
16 | funex 5327 | . . 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 391 | . 2 ⊢ (φ → {〈〈x, y〉, z〉 ∣ ((x ∈ A ∧ y ∈ B) ∧ ψ)} ∈ V) |
18 | 1, 17 | eqeltrd 2111 | 1 ⊢ (φ → 𝐹 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀wal 1240 = wceq 1242 ∈ wcel 1390 ∃*wmo 1898 Vcvv 2551 ⊆ wss 2911 × cxp 4286 dom cdm 4288 Fun wfun 4839 {coprab 5456 |
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-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-oprab 5459 |
This theorem is referenced by: (None) |
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