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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 5262 | . 2 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥) | |
2 | iotanul 5230 | . 2 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (℩𝑥𝐴𝐹𝑥) = ∅) | |
3 | 1, 2 | eqtrid 2238 | 1 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1364 ∃!weu 2042 ∅c0 3446 class class class wbr 4029 ℩cio 5213 ‘cfv 5254 |
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-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2175 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-dif 3155 df-in 3159 df-ss 3166 df-nul 3447 df-sn 3624 df-uni 3836 df-iota 5215 df-fv 5262 |
This theorem is referenced by: fvprc 5548 ndmfvg 5585 nfunsn 5589 |
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