Theorem tz6.12-2 5347

Description: Function value when F is not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by set.mm contributors, 30-Apr-2004.)

Assertion

Ref	Expression
tz6.12-2	|- (-. E!y AFy -> (F` A) = (/))

Distinct variable groups:   y,A   y,F

Proof of Theorem tz6.12-2

Dummy variables x z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fv3 5342	. 2 |- (F` A) = {x | (E.y(x e. y /\ AFy) /\ E!y AFy)}
2		vex 2863	. . . . . 6 |- z e. _V
3		elequ1 1713	. . . . . . . . 9 |- (x = z -> (x e. y <-> z e. y))
4	3	anbi1d 685	. . . . . . . 8 |- (x = z -> ((x e. y /\ AFy) <-> (z e. y /\ AFy)))
5	4	exbidv 1626	. . . . . . 7 |- (x = z -> (E.y(x e. y /\ AFy) <-> E.y(z e. y /\ AFy)))
6	5	anbi1d 685	. . . . . 6 |- (x = z -> ((E.y(x e. y /\ AFy) /\ E!y AFy) <-> (E.y(z e. y /\ AFy) /\ E!y AFy)))
7	2, 6	elab 2986	. . . . 5 |- (z e. {x | (E.y(x e. y /\ AFy) /\ E!y AFy)} <-> (E.y(z e. y /\ AFy) /\ E!y AFy))
8	7	simprbi 450	. . . 4 |- (z e. {x | (E.y(x e. y /\ AFy) /\ E!y AFy)} -> E!y AFy)
9	8	con3i 127	. . 3 |- (-. E!y AFy -> -. z e. {x | (E.y(x e. y /\ AFy) /\ E!y AFy)})
10	9	eq0rdv 3586	. 2 |- (-. E!y AFy -> {x | (E.y(x e. y /\ AFy) /\ E!y AFy)} = (/))
11	1, 10	syl5eq 2397	1 |- (-. E!y AFy -> (F` A) = (/))

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  E!weu 2204  {cab 2339  (/)c0 3551   class class class wbr 4640  ` cfv 4782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-br 4641  df-fv 4796

This theorem is referenced by:  tz6.12i  5349  ndmfv  5350  nfunsn  5354  fvfullfun  5865