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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 1411 | . 2 ⊢ ∀𝑥∀𝑦∃*𝑧𝜑 |
3 | funoprabg 5838 | . 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 1314 ∃*wmo 1978 Fun wfun 5087 {coprab 5743 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-fun 5095 df-oprab 5746 |
This theorem is referenced by: mpofun 5841 ovidig 5856 ovigg 5859 oprabex 5994 th3qcor 6501 axaddf 7644 axmulf 7645 |
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