Theorem tz6.12i 6214

Description: Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by Mario Carneiro, 17-Nov-2014.)

Assertion

Ref	Expression
tz6.12i	|- ( B =/= (/) -> ( ( F ` A ) = B -> A F B ) )

Proof of Theorem tz6.12i

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6201	. . . . 5 |- ( F ` A ) e. _V
2		neeq1 2856	. . . . . . . 8 |- ( ( F ` A ) = y -> ( ( F ` A ) =/= (/) <-> y =/= (/) ) )
3		tz6.12-2 6182	. . . . . . . . . . 11 |- ( -. E! y A F y -> ( F ` A ) = (/) )
4	3	necon1ai 2821	. . . . . . . . . 10 |- ( ( F ` A ) =/= (/) -> E! y A F y )
5		tz6.12c 6213	. . . . . . . . . 10 |- ( E! y A F y -> ( ( F ` A ) = y <-> A F y ) )
6	4, 5	syl 17	. . . . . . . . 9 |- ( ( F ` A ) =/= (/) -> ( ( F ` A ) = y <-> A F y ) )
7	6	biimpcd 239	. . . . . . . 8 |- ( ( F ` A ) = y -> ( ( F ` A ) =/= (/) -> A F y ) )
8	2, 7	sylbird 250	. . . . . . 7 |- ( ( F ` A ) = y -> ( y =/= (/) -> A F y ) )
9	8	eqcoms 2630	. . . . . 6 |- ( y = ( F ` A ) -> ( y =/= (/) -> A F y ) )
10		neeq1 2856	. . . . . 6 |- ( y = ( F ` A ) -> ( y =/= (/) <-> ( F ` A ) =/= (/) ) )
11		breq2 4657	. . . . . 6 |- ( y = ( F ` A ) -> ( A F y <-> A F ( F ` A ) ) )
12	9, 10, 11	3imtr3d 282	. . . . 5 |- ( y = ( F ` A ) -> ( ( F ` A ) =/= (/) -> A F ( F ` A ) ) )
13	1, 12	vtocle 3282	. . . 4 |- ( ( F ` A ) =/= (/) -> A F ( F ` A ) )
14	13	a1i 11	. . 3 |- ( ( F ` A ) = B -> ( ( F ` A ) =/= (/) -> A F ( F ` A ) ) )
15		neeq1 2856	. . 3 |- ( ( F ` A ) = B -> ( ( F ` A ) =/= (/) <-> B =/= (/) ) )
16		breq2 4657	. . 3 |- ( ( F ` A ) = B -> ( A F ( F ` A ) <-> A F B ) )
17	14, 15, 16	3imtr3d 282	. 2 |- ( ( F ` A ) = B -> ( B =/= (/) -> A F B ) )
18	17	com12 32	1 |- ( B =/= (/) -> ( ( F ` A ) = B -> A F B ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   E!weu 2470   =/= wne 2794   (/)c0 3915   class class class wbr 4653   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  fvbr0  6215  fvclss  6500  dcomex  9269