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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 5778 | . . 3 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} | |
2 | nfopab1 5176 | . . . 4 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑} | |
3 | nfopab2 5177 | . . . 4 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑} | |
4 | 2, 3 | dffun6f 6515 | . . 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 5107 | . . . . 5 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) | |
7 | opabidw 5482 | . . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑) | |
8 | 6, 7 | bitri 275 | . . . 4 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ 𝜑) |
9 | 8 | mobii 2543 | . . 3 ⊢ (∃*𝑦 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ∃*𝑦𝜑) |
10 | 9 | albii 1822 | . 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 205 ∀wal 1540 ∈ wcel 2107 ∃*wmo 2533 ⟨cop 4593 class class class wbr 5106 {copab 5168 Rel wrel 5639 Fun wfun 6491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-fun 6499 |
This theorem is referenced by: funopabeq 6538 funco 6542 isarep2 6593 mptfnf 6637 fnopabg 6639 opabiotafun 6923 fvopab3ig 6945 opabex 7171 funoprabg 7478 zfrep6 7888 tz7.44lem1 8352 pwfir 9123 ajfuni 29843 funadj 30870 abrexdomjm 31476 fineqvrep 33753 satfv0fun 34022 satffunlem1lem1 34053 satffunlem2lem1 34055 abrexdom 36235 |
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