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Mirrors > Home > ILE Home > Th. List > tz6.12-2 | GIF version |
Description: Function value when 𝐹 is not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
tz6.12-2 | ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fv 5131 | . 2 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥) | |
2 | iotanul 5103 | . 2 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (℩𝑥𝐴𝐹𝑥) = ∅) | |
3 | 1, 2 | syl5eq 2184 | 1 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1331 ∃!weu 1999 ∅c0 3363 class class class wbr 3929 ℩cio 5086 ‘cfv 5123 |
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-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-in 3077 df-ss 3084 df-nul 3364 df-sn 3533 df-uni 3737 df-iota 5088 df-fv 5131 |
This theorem is referenced by: fvprc 5415 ndmfvg 5452 nfunsn 5455 |
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