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Mirrors > Home > MPE Home > Th. List > funoprab | Structured version Visualization version GIF version |
Description: "At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 17-Mar-1995.) |
Ref | Expression |
---|---|
funoprab.1 | ⊢ ∃*𝑧𝜑 |
Ref | Expression |
---|---|
funoprab | ⊢ Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funoprab.1 | . . 3 ⊢ ∃*𝑧𝜑 | |
2 | 1 | gen2 1790 | . 2 ⊢ ∀𝑥∀𝑦∃*𝑧𝜑 |
3 | funoprabg 7521 | . 2 ⊢ (∀𝑥∀𝑦∃*𝑧𝜑 → Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∀wal 1531 ∃*wmo 2524 Fun wfun 6527 {coprab 7402 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-br 5139 df-opab 5201 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-fun 6535 df-oprab 7405 |
This theorem is referenced by: mpofun 7524 mpofunOLD 7525 ovidig 7542 ovigg 7545 oprabex 7956 axaddf 11136 axmulf 11137 funtransport 35498 funray 35607 funline 35609 |
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