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	⊢ (¬ ∃!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 ∣ (∃y(x ∈ y ∧ AFy) ∧ ∃!y AFy)}
2		vex 2863	. . . . . 6 ⊢ z ∈ V
3		elequ1 1713	. . . . . . . . 9 ⊢ (x = z → (x ∈ y ↔ z ∈ y))
4	3	anbi1d 685	. . . . . . . 8 ⊢ (x = z → ((x ∈ y ∧ AFy) ↔ (z ∈ y ∧ AFy)))
5	4	exbidv 1626	. . . . . . 7 ⊢ (x = z → (∃y(x ∈ y ∧ AFy) ↔ ∃y(z ∈ y ∧ AFy)))
6	5	anbi1d 685	. . . . . 6 ⊢ (x = z → ((∃y(x ∈ y ∧ AFy) ∧ ∃!y AFy) ↔ (∃y(z ∈ y ∧ AFy) ∧ ∃!y AFy)))
7	2, 6	elab 2986	. . . . 5 ⊢ (z ∈ {x ∣ (∃y(x ∈ y ∧ AFy) ∧ ∃!y AFy)} ↔ (∃y(z ∈ y ∧ AFy) ∧ ∃!y AFy))
8	7	simprbi 450	. . . 4 ⊢ (z ∈ {x ∣ (∃y(x ∈ y ∧ AFy) ∧ ∃!y AFy)} → ∃!y AFy)
9	8	con3i 127	. . 3 ⊢ (¬ ∃!y AFy → ¬ z ∈ {x ∣ (∃y(x ∈ y ∧ AFy) ∧ ∃!y AFy)})
10	9	eq0rdv 3586	. 2 ⊢ (¬ ∃!y AFy → {x ∣ (∃y(x ∈ y ∧ AFy) ∧ ∃!y AFy)} = ∅)
11	1, 10	syl5eq 2397	1 ⊢ (¬ ∃!y AFy → (F ‘A) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!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