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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 | relopab 5660 | . . 3 ⊢ Rel {〈𝑥, 𝑦〉 ∣ 𝜑} | |
2 | nfopab1 5099 | . . . 4 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} | |
3 | nfopab2 5100 | . . . 4 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} | |
4 | 2, 3 | dffun6f 6338 | . . 3 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ (Rel {〈𝑥, 𝑦〉 ∣ 𝜑} ∧ ∀𝑥∃*𝑦 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦)) |
5 | 1, 4 | mpbiran 708 | . 2 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∀𝑥∃*𝑦 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦) |
6 | df-br 5031 | . . . . 5 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
7 | opabidw 5377 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | |
8 | 6, 7 | bitri 278 | . . . 4 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 𝜑) |
9 | 8 | mobii 2606 | . . 3 ⊢ (∃*𝑦 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ ∃*𝑦𝜑) |
10 | 9 | albii 1821 | . 2 ⊢ (∀𝑥∃*𝑦 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ ∀𝑥∃*𝑦𝜑) |
11 | 5, 10 | bitri 278 | 1 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∀𝑥∃*𝑦𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∀wal 1536 ∈ wcel 2111 ∃*wmo 2596 〈cop 4531 class class class wbr 5030 {copab 5092 Rel wrel 5524 Fun wfun 6318 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-fun 6326 |
This theorem is referenced by: funopabeq 6360 funco 6364 isarep2 6413 mptfnf 6455 fnopabg 6457 opabiotafun 6719 fvopab3ig 6741 opabex 6960 funoprabg 7252 zfrep6 7638 tz7.44lem1 8024 ajfuni 28642 funadj 29669 abrexdomjm 30275 satfv0fun 32731 satffunlem1lem1 32762 satffunlem2lem1 32764 abrexdom 35168 |
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