Theorem ralrn 5703

Description: Restricted universal quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)

Hypothesis

Ref	Expression
rexrn.1	|- ( x = ( F ` y ) -> ( ph <-> ps ) )

Assertion

Ref	Expression
ralrn	|- ( F Fn A -> ( A. x e. ran F ph <-> A. y e. A ps ) )

Distinct variable groups:   x, y, A   x, F, y   ps, x   ph, y

Allowed substitution hints:   ph( x)   ps( y)

Proof of Theorem ralrn

Step	Hyp	Ref	Expression
1		funfvex 5578	. . 3 |- ( ( Fun F /\ y e. dom F ) -> ( F ` y ) e. _V )
2	1	funfni 5361	. 2 |- ( ( F Fn A /\ y e. A ) -> ( F ` y ) e. _V )
3		fvelrnb 5611	. . 3 |- ( F Fn A -> ( x e. ran F <-> E. y e. A ( F ` y ) = x ) )
4		eqcom 2198	. . . 4 |- ( ( F ` y ) = x <-> x = ( F ` y ) )
5	4	rexbii 2504	. . 3 |- ( E. y e. A ( F ` y ) = x <-> E. y e. A x = ( F ` y ) )
6	3, 5	bitrdi 196	. 2 |- ( F Fn A -> ( x e. ran F <-> E. y e. A x = ( F ` y ) ) )
7		rexrn.1	. . 3 |- ( x = ( F ` y ) -> ( ph <-> ps ) )
8	7	adantl 277	. 2 |- ( ( F Fn A /\ x = ( F ` y ) ) -> ( ph <-> ps ) )
9	2, 6, 8	ralxfr2d 4500	1 |- ( F Fn A -> ( A. x e. ran F ph <-> A. y e. A ps ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2167   A.wral 2475   E.wrex 2476   _Vcvv 2763   ran crn 4665   Fn wfn 5254   ` cfv 5259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267

This theorem is referenced by:  ralrnmpt  5707  cbvfo  5835  isoselem  5870  difinfsn  7175  imasaddfnlemg  13016  ghmrn  13463  ghmnsgima  13474  znf1o  14283  nninfall  15740