Theorem tz6.12c 2846

Description: Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27.

Hypothesis

Ref	Expression
tz6.12c.1	⊢ A ∈ V

Assertion

Ref	Expression
tz6.12c	⊢ (∃!y AFy → ((F ‘A) = y ↔ AFy))

Distinct variable group(s):   y,F   y,A

Proof of Theorem tz6.12c

Step	Hyp	Ref	Expression
1		euex 1021	. . . 4 ⊢ (∃!y AFy → ∃y AFy)
2		hbeu1 1015	. . . . . 6 ⊢ (∃!y AFy → ∀y∃!y AFy)
3		ax-17 925	. . . . . 6 ⊢ (AF(F ‘A) → ∀y AF(F ‘A))
4	2, 3	hbim 702	. . . . 5 ⊢ ((∃!y AFy → AF(F ‘A)) → ∀y(∃!y AFy → AF(F ‘A)))
5		tz6.12c.1	. . . . . . . . . 10 ⊢ A ∈ V
6	5	tz6.12-1 2842	. . . . . . . . 9 ⊢ ((AFy ∧ ∃!y AFy) → (F ‘A) = y)
7	6	exp 291	. . . . . . . 8 ⊢ (AFy → (∃!y AFy → (F ‘A) = y))
8	7	com12 13	. . . . . . 7 ⊢ (∃!y AFy → (AFy → (F ‘A) = y))
9		breq2 2066	. . . . . . . 8 ⊢ ((F ‘A) = y → (AF(F ‘A) ↔ AFy))
10	9	biimprd 136	. . . . . . 7 ⊢ ((F ‘A) = y → (AFy → AF(F ‘A)))
11	8, 10	syli 52	. . . . . 6 ⊢ (∃!y AFy → (AFy → AF(F ‘A)))
12	11	com12 13	. . . . 5 ⊢ (AFy → (∃!y AFy → AF(F ‘A)))
13	4, 12	19.23ai 746	. . . 4 ⊢ (∃y AFy → (∃!y AFy → AF(F ‘A)))
14	1, 13	mpcom 49	. . 3 ⊢ (∃!y AFy → AF(F ‘A))
15	9	biimpcd 137	. . 3 ⊢ (AF(F ‘A) → ((F ‘A) = y → AFy))
16	14, 15	syl 12	. 2 ⊢ (∃!y AFy → ((F ‘A) = y → AFy))
17	16, 8	impbid 397	1 ⊢ (∃!y AFy → ((F ‘A) = y ↔ AFy))

Syntax hints:   → wi 2   ↔ wb 127  ∃wex 678  ∃!weu 1007   = wceq 1091   ∈ wcel 1092  Vcvv 1348   class class class wbr 2054   ‘cfv 2422

This theorem is referenced by:  tz6.12i 2847  fnfvbr 2855

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-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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