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Mirrors > Home > ILE Home > Th. List > funoprab | 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 1450 | . 2 ⊢ ∀𝑥∀𝑦∃*𝑧𝜑 |
3 | funoprabg 5967 | . 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 set class |
Syntax hints: ∀wal 1351 ∃*wmo 2027 Fun wfun 5205 {coprab 5869 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 df-opab 4062 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-fun 5213 df-oprab 5872 |
This theorem is referenced by: mpofun 5970 ovidig 5985 ovigg 5988 oprabex 6122 th3qcor 6632 axaddf 7845 axmulf 7846 |
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