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Mirrors > Home > ILE Home > Th. List > opabex | GIF version |
Description: Existence of a function expressed as class of ordered pairs. (Contributed by NM, 21-Jul-1996.) |
Ref | Expression |
---|---|
opabex.1 | ⊢ 𝐴 ∈ V |
opabex.2 | ⊢ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑) |
Ref | Expression |
---|---|
opabex | ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funopab 5263 | . . 3 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | opabex.2 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑) | |
3 | moanimv 2111 | . . . 4 ⊢ (∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑)) | |
4 | 2, 3 | mpbir 146 | . . 3 ⊢ ∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) |
5 | 1, 4 | mpgbir 1463 | . 2 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
6 | opabex.1 | . . 3 ⊢ 𝐴 ∈ V | |
7 | dmopabss 4851 | . . 3 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 | |
8 | 6, 7 | ssexi 4153 | . 2 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V |
9 | funex 5752 | . 2 ⊢ ((Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V) | |
10 | 5, 8, 9 | mp2an 426 | 1 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∃*wmo 2037 ∈ wcel 2158 Vcvv 2749 {copab 4075 dom cdm 4638 Fun wfun 5222 |
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 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-pow 4186 ax-pr 4221 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 |
This theorem is referenced by: (None) |
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