Theorem tz6.12c 5348

Description: Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)

Assertion

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

Distinct variable groups:   y,F   y,A

Allowed substitution hint:   B(y)

Proof of Theorem tz6.12c

Step	Hyp	Ref	Expression
1		euex 2227	. . . 4 ⊢ (∃!y AFy → ∃y AFy)
2		nfeu1 2214	. . . . . 6 ⊢ Ⅎy∃!y AFy
3		nfv 1619	. . . . . 6 ⊢ Ⅎy AF(F ‘A)
4	2, 3	nfim 1813	. . . . 5 ⊢ Ⅎy(∃!y AFy → AF(F ‘A))
5		tz6.12-1 5345	. . . . . . . 8 ⊢ ((AFy ∧ ∃!y AFy) → (F ‘A) = y)
6	5	expcom 424	. . . . . . 7 ⊢ (∃!y AFy → (AFy → (F ‘A) = y))
7		breq2 4644	. . . . . . . 8 ⊢ ((F ‘A) = y → (AF(F ‘A) ↔ AFy))
8	7	biimprd 214	. . . . . . 7 ⊢ ((F ‘A) = y → (AFy → AF(F ‘A)))
9	6, 8	syli 33	. . . . . 6 ⊢ (∃!y AFy → (AFy → AF(F ‘A)))
10	9	com12 27	. . . . 5 ⊢ (AFy → (∃!y AFy → AF(F ‘A)))
11	4, 10	exlimi 1803	. . . 4 ⊢ (∃y AFy → (∃!y AFy → AF(F ‘A)))
12	1, 11	mpcom 32	. . 3 ⊢ (∃!y AFy → AF(F ‘A))
13		breq2 4644	. . 3 ⊢ ((F ‘A) = B → (AF(F ‘A) ↔ AFB))
14	12, 13	syl5ibcom 211	. 2 ⊢ (∃!y AFy → ((F ‘A) = B → AFB))
15		tz6.12-1 5345	. . 3 ⊢ ((AFB ∧ ∃!y AFy) → (F ‘A) = B)
16	15	expcom 424	. 2 ⊢ (∃!y AFy → (AFB → (F ‘A) = B))
17	14, 16	impbid 183	1 ⊢ (∃!y AFy → ((F ‘A) = B ↔ AFB))

Syntax hints:   → wi 4   ↔ wb 176  ∃wex 1541   = wceq 1642  ∃!weu 2204   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-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  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-pss 3262  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-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-br 4641  df-fv 4796

This theorem is referenced by:  tz6.12i  5349  fnbrfvb  5359