Theorem bnj1542 30927

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1542.1	|- ( ph -> F Fn A )
bnj1542.2	|- ( ph -> G Fn A )
bnj1542.3	|- ( ph -> F =/= G )
bnj1542.4	|- ( w e. F -> A. x w e. F )

Assertion

Ref	Expression
bnj1542	|- ( ph -> E. x e. A ( F ` x ) =/= ( G ` x ) )

Distinct variable groups:   x, A   w, F   w, G, x

Allowed substitution hints:   ph( x, w)   A( w)   F( x)

Proof of Theorem bnj1542

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1542.3	. . 3 |- ( ph -> F =/= G )
2		bnj1542.1	. . . 4 |- ( ph -> F Fn A )
3		bnj1542.2	. . . 4 |- ( ph -> G Fn A )
4		eqfnfv 6311	. . . . . 6 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> A. y e. A ( F ` y ) = ( G ` y ) ) )
5	4	necon3abid 2830	. . . . 5 |- ( ( F Fn A /\ G Fn A ) -> ( F =/= G <-> -. A. y e. A ( F ` y ) = ( G ` y ) ) )
6		df-ne 2795	. . . . . . 7 |- ( ( F ` y ) =/= ( G ` y ) <-> -. ( F ` y ) = ( G ` y ) )
7	6	rexbii 3041	. . . . . 6 |- ( E. y e. A ( F ` y ) =/= ( G ` y ) <-> E. y e. A -. ( F ` y ) = ( G ` y ) )
8		rexnal 2995	. . . . . 6 |- ( E. y e. A -. ( F ` y ) = ( G ` y ) <-> -. A. y e. A ( F ` y ) = ( G ` y ) )
9	7, 8	bitri 264	. . . . 5 |- ( E. y e. A ( F ` y ) =/= ( G ` y ) <-> -. A. y e. A ( F ` y ) = ( G ` y ) )
10	5, 9	syl6bbr 278	. . . 4 |- ( ( F Fn A /\ G Fn A ) -> ( F =/= G <-> E. y e. A ( F ` y ) =/= ( G ` y ) ) )
11	2, 3, 10	syl2anc 693	. . 3 |- ( ph -> ( F =/= G <-> E. y e. A ( F ` y ) =/= ( G ` y ) ) )
12	1, 11	mpbid 222	. 2 |- ( ph -> E. y e. A ( F ` y ) =/= ( G ` y ) )
13		nfv 1843	. . 3 |- F/ y ( F ` x ) =/= ( G ` x )
14		bnj1542.4	. . . . . 6 |- ( w e. F -> A. x w e. F )
15	14	nfcii 2755	. . . . 5 |- F/_ x F
16		nfcv 2764	. . . . 5 |- F/_ x y
17	15, 16	nffv 6198	. . . 4 |- F/_ x ( F ` y )
18		nfcv 2764	. . . 4 |- F/_ x ( G ` y )
19	17, 18	nfne 2894	. . 3 |- F/ x ( F ` y ) =/= ( G ` y )
20		fveq2 6191	. . . 4 |- ( x = y -> ( F ` x ) = ( F ` y ) )
21		fveq2 6191	. . . 4 |- ( x = y -> ( G ` x ) = ( G ` y ) )
22	20, 21	neeq12d 2855	. . 3 |- ( x = y -> ( ( F ` x ) =/= ( G ` x ) <-> ( F ` y ) =/= ( G ` y ) ) )
23	13, 19, 22	cbvrex 3168	. 2 |- ( E. x e. A ( F ` x ) =/= ( G ` x ) <-> E. y e. A ( F ` y ) =/= ( G ` y ) )
24	12, 23	sylibr 224	1 |- ( ph -> E. x e. A ( F ` x ) =/= ( G ` x ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   Fn wfn 5883   ` 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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

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-mo 2475  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-csb 3534  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-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  bnj1523  31139