Theorem ralrn 5696

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 5571	. . 3 |- ( ( Fun F /\ y e. dom F ) -> ( F ` y ) e. _V )
2	1	funfni 5354	. 2 |- ( ( F Fn A /\ y e. A ) -> ( F ` y ) e. _V )
3		fvelrnb 5604	. . 3 |- ( F Fn A -> ( x e. ran F <-> E. y e. A ( F ` y ) = x ) )
4		eqcom 2195	. . . 4 |- ( ( F ` y ) = x <-> x = ( F ` y ) )
5	4	rexbii 2501	. . 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 4495	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 2164   A.wral 2472   E.wrex 2473   _Vcvv 2760   ran crn 4660   Fn wfn 5249   ` cfv 5254

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262

This theorem is referenced by:  ralrnmpt  5700  cbvfo  5828  isoselem  5863  difinfsn  7159  imasaddfnlemg  12897  ghmrn  13327  ghmnsgima  13338  znf1o  14139  nninfall  15499