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| Mirrors > Home > MPE Home > Th. List > tz6.12-2 | Structured version Visualization version 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 6569 | . 2 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥) | |
| 2 | iotanul 6539 | . 2 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (℩𝑥𝐴𝐹𝑥) = ∅) | |
| 3 | 1, 2 | eqtrid 2789 | 1 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∃!weu 2568 ∅c0 4333 class class class wbr 5143 ℩cio 6512 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-v 3482 df-dif 3954 df-ss 3968 df-nul 4334 df-sn 4627 df-uni 4908 df-iota 6514 df-fv 6569 |
| This theorem is referenced by: fvprc 6898 fvprcALT 6899 tz6.12i 6934 ndmfv 6941 nfunsn 6948 funpartfv 35946 setrec2lem1 49212 |
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