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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 6570 | . 2 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥) | |
2 | iotanul 6540 | . 2 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (℩𝑥𝐴𝐹𝑥) = ∅) | |
3 | 1, 2 | eqtrid 2786 | 1 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1536 ∃!weu 2565 ∅c0 4338 class class class wbr 5147 ℩cio 6513 ‘cfv 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-v 3479 df-dif 3965 df-ss 3979 df-nul 4339 df-sn 4631 df-uni 4912 df-iota 6515 df-fv 6570 |
This theorem is referenced by: fvprc 6898 fvprcALT 6899 tz6.12i 6934 ndmfv 6941 nfunsn 6948 funpartfv 35926 setrec2lem1 48923 |
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