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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 5691 | . . 3 ⊢ Rel {〈𝑥, 𝑦〉 ∣ 𝜑} | |
2 | nfopab1 5128 | . . . 4 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} | |
3 | nfopab2 5129 | . . . 4 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} | |
4 | 2, 3 | dffun6f 6364 | . . 3 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ (Rel {〈𝑥, 𝑦〉 ∣ 𝜑} ∧ ∀𝑥∃*𝑦 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦)) |
5 | 1, 4 | mpbiran 707 | . 2 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∀𝑥∃*𝑦 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦) |
6 | df-br 5060 | . . . . 5 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
7 | opabidw 5405 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | |
8 | 6, 7 | bitri 277 | . . . 4 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 𝜑) |
9 | 8 | mobii 2627 | . . 3 ⊢ (∃*𝑦 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ ∃*𝑦𝜑) |
10 | 9 | albii 1816 | . 2 ⊢ (∀𝑥∃*𝑦 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ ∀𝑥∃*𝑦𝜑) |
11 | 5, 10 | bitri 277 | 1 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∀𝑥∃*𝑦𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∀wal 1531 ∈ wcel 2110 ∃*wmo 2616 〈cop 4567 class class class wbr 5059 {copab 5121 Rel wrel 5555 Fun wfun 6344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-fun 6352 |
This theorem is referenced by: funopabeq 6386 funco 6390 isarep2 6438 mptfnf 6478 fnopabg 6480 opabiotafun 6739 fvopab3ig 6759 opabex 6977 funoprabg 7267 zfrep6 7650 tz7.44lem1 8035 ajfuni 28630 funadj 29657 abrexdomjm 30261 satfv0fun 32613 satffunlem1lem1 32644 satffunlem2lem1 32646 abrexdom 34999 |
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