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Mirrors > Home > ILE Home > Th. List > funopab | GIF version |
Description: A class of ordered pairs is a function when there is at most one second member for each pair. (Contributed by NM, 16-May-1995.) |
Ref | Expression |
---|---|
funopab | ⊢ (Fun {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∀𝑥∃*𝑦𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relopab 4482 | . . 3 ⊢ Rel {〈𝑥, 𝑦〉 ∣ 𝜑} | |
2 | nfopab1 3847 | . . . 4 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} | |
3 | nfopab2 3848 | . . . 4 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} | |
4 | 2, 3 | dffun6f 4935 | . . 3 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ (Rel {〈𝑥, 𝑦〉 ∣ 𝜑} ∧ ∀𝑥∃*𝑦 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦)) |
5 | 1, 4 | mpbiran 881 | . 2 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∀𝑥∃*𝑦 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦) |
6 | df-br 3786 | . . . . 5 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
7 | opabid 4012 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | |
8 | 6, 7 | bitri 182 | . . . 4 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 𝜑) |
9 | 8 | mobii 1978 | . . 3 ⊢ (∃*𝑦 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ ∃*𝑦𝜑) |
10 | 9 | albii 1399 | . 2 ⊢ (∀𝑥∃*𝑦 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ ∀𝑥∃*𝑦𝜑) |
11 | 5, 10 | bitri 182 | 1 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∀𝑥∃*𝑦𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 ∀wal 1282 ∈ wcel 1433 ∃*wmo 1942 〈cop 3401 class class class wbr 3785 {copab 3838 Rel wrel 4368 Fun wfun 4916 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-fun 4924 |
This theorem is referenced by: funopabeq 4956 isarep2 5006 fnopabg 5042 fvopab3ig 5267 opabex 5406 funoprabg 5620 |
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