Theorem tz6.12-2 5617

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.)

Assertion

Ref	Expression
tz6.12-2	⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅)

Distinct variable groups:   𝑥,𝐹   𝑥,𝐴

Proof of Theorem tz6.12-2

Step	Hyp	Ref	Expression
1		df-fv 5325	. 2 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
2		iotanul 5293	. 2 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (℩𝑥𝐴𝐹𝑥) = ∅)
3	1, 2	eqtrid 2274	1 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1395  ∃!weu 2077  ∅c0 3491   class class class wbr 4082  ℩cio 5275  ‘cfv 5317

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-in 3203  df-ss 3210  df-nul 3492  df-sn 3672  df-uni 3888  df-iota 5277  df-fv 5325

This theorem is referenced by:  fvprc  5620  ndmfvg  5657  nfunsn  5663