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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 4577 | . . 3 ⊢ Rel {〈𝑥, 𝑦〉 ∣ 𝜑} | |
2 | nfopab1 3913 | . . . 4 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} | |
3 | nfopab2 3914 | . . . 4 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} | |
4 | 2, 3 | dffun6f 5041 | . . 3 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ (Rel {〈𝑥, 𝑦〉 ∣ 𝜑} ∧ ∀𝑥∃*𝑦 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦)) |
5 | 1, 4 | mpbiran 887 | . 2 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∀𝑥∃*𝑦 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦) |
6 | df-br 3852 | . . . . 5 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
7 | opabid 4093 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | |
8 | 6, 7 | bitri 183 | . . . 4 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 𝜑) |
9 | 8 | mobii 1986 | . . 3 ⊢ (∃*𝑦 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ ∃*𝑦𝜑) |
10 | 9 | albii 1405 | . 2 ⊢ (∀𝑥∃*𝑦 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ ∀𝑥∃*𝑦𝜑) |
11 | 5, 10 | bitri 183 | 1 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∀𝑥∃*𝑦𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∀wal 1288 ∈ wcel 1439 ∃*wmo 1950 〈cop 3453 class class class wbr 3851 {copab 3904 Rel wrel 4457 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-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-v 2622 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-br 3852 df-opab 3906 df-id 4129 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-fun 5030 |
This theorem is referenced by: funopabeq 5063 isarep2 5114 fnopabg 5150 fvopab3ig 5391 opabex 5535 funoprabg 5758 |
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