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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 5062 | . . 3 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | opabex.2 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑) | |
3 | moanimv 2024 | . . . 4 ⊢ (∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑)) | |
4 | 2, 3 | mpbir 145 | . . 3 ⊢ ∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) |
5 | 1, 4 | mpgbir 1388 | . 2 ⊢ Fun {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
6 | opabex.1 | . . 3 ⊢ 𝐴 ∈ V | |
7 | dmopabss 4661 | . . 3 ⊢ dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 | |
8 | 6, 7 | ssexi 3983 | . 2 ⊢ dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V |
9 | funex 5534 | . 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 418 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1439 ∃*wmo 1950 Vcvv 2620 {copab 3904 dom cdm 4452 Fun wfun 5022 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-pow 4015 ax-pr 4045 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-id 4129 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 |
This theorem is referenced by: (None) |
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