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| Mirrors > Home > MPE Home > Th. List > funopab | Structured version Visualization version 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 | relopabv 5831 | . . 3 ⊢ Rel {〈𝑥, 𝑦〉 ∣ 𝜑} | |
| 2 | nfopab1 5213 | . . . 4 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} | |
| 3 | nfopab2 5214 | . . . 4 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} | |
| 4 | 2, 3 | dffun6f 6579 | . . 3 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ (Rel {〈𝑥, 𝑦〉 ∣ 𝜑} ∧ ∀𝑥∃*𝑦 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦)) |
| 5 | 1, 4 | mpbiran 709 | . 2 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∀𝑥∃*𝑦 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦) |
| 6 | df-br 5144 | . . . . 5 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
| 7 | opabidw 5529 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | |
| 8 | 6, 7 | bitri 275 | . . . 4 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 𝜑) |
| 9 | 8 | mobii 2548 | . . 3 ⊢ (∃*𝑦 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ ∃*𝑦𝜑) |
| 10 | 9 | albii 1819 | . 2 ⊢ (∀𝑥∃*𝑦 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ ∀𝑥∃*𝑦𝜑) |
| 11 | 5, 10 | bitri 275 | 1 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∀𝑥∃*𝑦𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∀wal 1538 ∈ wcel 2108 ∃*wmo 2538 〈cop 4632 class class class wbr 5143 {copab 5205 Rel wrel 5690 Fun wfun 6555 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-fun 6563 |
| This theorem is referenced by: funopabeq 6602 funco 6606 isarep2 6658 mptfnf 6703 fnopabg 6705 opabiotafun 6989 fvopab3ig 7012 opabex 7240 funoprabg 7554 zfrep6 7979 tz7.44lem1 8445 pwfir 9355 ajfuni 30878 funadj 31905 abrexdomjm 32526 fineqvrep 35109 satfv0fun 35376 satffunlem1lem1 35407 satffunlem2lem1 35409 abrexdom 37737 modelaxreplem2 44996 |
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