Theorem tz6.12i 2847

Description: Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27.

Hypothesis

Ref	Expression
tz6.12i.1	⊢ A ∈ V

Assertion

Ref	Expression
tz6.12i	⊢ (¬ B = ∅ → ((F ‘A) = B → AFB))

Proof of Theorem tz6.12i

Step	Hyp	Ref	Expression
1		fvex 2838	. . . 4 ⊢ (F ‘A) ∈ V
2		eleq1 1149	. . . 4 ⊢ ((F ‘A) = B → ((F ‘A) ∈ V ↔ B ∈ V))
3	1, 2	mpbii 168	. . 3 ⊢ ((F ‘A) = B → B ∈ V)
4		cleq2 1110	. . . . 5 ⊢ (y = B → ((F ‘A) = y ↔ (F ‘A) = B))
5		cleq1 1107	. . . . . . 7 ⊢ (y = B → (y = ∅ ↔ B = ∅))
6	5	negbid 463	. . . . . 6 ⊢ (y = B → (¬ y = ∅ ↔ ¬ B = ∅))
7		breq2 2066	. . . . . 6 ⊢ (y = B → (AFy ↔ AFB))
8	6, 7	imbi12d 474	. . . . 5 ⊢ (y = B → ((¬ y = ∅ → AFy) ↔ (¬ B = ∅ → AFB)))
9	4, 8	imbi12d 474	. . . 4 ⊢ (y = B → (((F ‘A) = y → (¬ y = ∅ → AFy)) ↔ ((F ‘A) = B → (¬ B = ∅ → AFB))))
10		cleq1 1107	. . . . . . 7 ⊢ ((F ‘A) = y → ((F ‘A) = ∅ ↔ y = ∅))
11	10	negbid 463	. . . . . 6 ⊢ ((F ‘A) = y → (¬ (F ‘A) = ∅ ↔ ¬ y = ∅))
12		tz6.12i.1	. . . . . . . . 9 ⊢ A ∈ V
13	12	tz6.12c 2846	. . . . . . . 8 ⊢ (∃!y AFy → ((F ‘A) = y ↔ AFy))
14		tz6.12-2 2845	. . . . . . . 8 ⊢ (¬ ∃!y AFy → (F ‘A) = ∅)
15	13, 14	nsyl4 105	. . . . . . 7 ⊢ (¬ (F ‘A) = ∅ → ((F ‘A) = y ↔ AFy))
16	15	biimpd 135	. . . . . 6 ⊢ (¬ (F ‘A) = ∅ → ((F ‘A) = y → AFy))
17	11, 16	syl6bir 188	. . . . 5 ⊢ ((F ‘A) = y → (¬ y = ∅ → ((F ‘A) = y → AFy)))
18	17	pm2.43a 60	. . . 4 ⊢ ((F ‘A) = y → (¬ y = ∅ → AFy))
19	9, 18	vtoclg 1383	. . 3 ⊢ (B ∈ V → ((F ‘A) = B → (¬ B = ∅ → AFB)))
20	3, 19	mpcom 49	. 2 ⊢ ((F ‘A) = B → (¬ B = ∅ → AFB))
21	20	com12 13	1 ⊢ (¬ B = ∅ → ((F ‘A) = B → AFB))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127  ∃!weu 1007   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ∅c0 1707   class class class wbr 2054   ‘cfv 2422

This theorem is referenced by:  fvclss 2907

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438

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