Theorem rdgeq2 7508

Description: Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)

Assertion

Ref	Expression
rdgeq2	|- ( A = B -> rec ( F , A ) = rec ( F , B ) )

Proof of Theorem rdgeq2

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ifeq1 4090	. . . 4 |- ( A = B -> if ( g = (/) , A , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) = if ( g = (/) , B , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) )
2	1	mpteq2dv 4745	. . 3 |- ( A = B -> ( g e. _V |-> if ( g = (/) , A , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) ) = ( g e. _V |-> if ( g = (/) , B , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) ) )
3		recseq 7470	. . 3 |- ( ( g e. _V |-> if ( g = (/) , A , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) ) = ( g e. _V |-> if ( g = (/) , B , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) ) -> recs ( ( g e. _V |-> if ( g = (/) , A , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) ) ) = recs ( ( g e. _V |-> if ( g = (/) , B , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) ) ) )
4	2, 3	syl 17	. 2 |- ( A = B -> recs ( ( g e. _V |-> if ( g = (/) , A , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) ) ) = recs ( ( g e. _V |-> if ( g = (/) , B , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) ) ) )
5		df-rdg 7506	. 2 |- rec ( F , A ) = recs ( ( g e. _V |-> if ( g = (/) , A , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) ) )
6		df-rdg 7506	. 2 |- rec ( F , B ) = recs ( ( g e. _V |-> if ( g = (/) , B , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) ) )
7	4, 5, 6	3eqtr4g 2681	1 |- ( A = B -> rec ( F , A ) = rec ( F , B ) )

Syntax hints:   -> wi 4   = wceq 1483   _Vcvv 3200   (/)c0 3915   ifcif 4086   U.cuni 4436   |-> cmpt 4729   dom cdm 5114   ran crn 5115   Lim wlim 5724   ` cfv 5888  recscrecs 7467   reccrdg 7505

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

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  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-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-fv 5896  df-wrecs 7407  df-recs 7468  df-rdg 7506

This theorem is referenced by:  rdgeq12  7509  rdg0g  7523  oav  7591  itunifval  9238  hsmex  9254  ltweuz  12760  seqeq1  12804  dfrdg2  31701  trpredeq3  31722  finxpeq2  33224  finxpreclem6  33233  finxpsuclem  33234