Theorem seqomeq12 7549

Description: Equality theorem for seq_𝜔. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Assertion

Ref	Expression
seqomeq12	|- ( ( A = B /\ C = D ) -> seq𝜔 ( A , C ) = seq𝜔 ( B , D ) )

Proof of Theorem seqomeq12

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq 6656	. . . . . 6 |- ( A = B -> ( a A b ) = ( a B b ) )
2	1	opeq2d 4409	. . . . 5 |- ( A = B -> <. suc a , ( a A b ) >. = <. suc a , ( a B b ) >. )
3	2	mpt2eq3dv 6721	. . . 4 |- ( A = B -> ( a e. om , b e. _V |-> <. suc a , ( a A b ) >. ) = ( a e. om , b e. _V |-> <. suc a , ( a B b ) >. ) )
4		fveq2 6191	. . . . 5 |- ( C = D -> ( _I ` C ) = ( _I ` D ) )
5	4	opeq2d 4409	. . . 4 |- ( C = D -> <. (/) , ( _I ` C ) >. = <. (/) , ( _I ` D ) >. )
6		rdgeq12 7509	. . . 4 |- ( ( ( a e. om , b e. _V |-> <. suc a , ( a A b ) >. ) = ( a e. om , b e. _V |-> <. suc a , ( a B b ) >. ) /\ <. (/) , ( _I ` C ) >. = <. (/) , ( _I ` D ) >. ) -> rec ( ( a e. om , b e. _V |-> <. suc a , ( a A b ) >. ) , <. (/) , ( _I ` C ) >. ) = rec ( ( a e. om , b e. _V |-> <. suc a , ( a B b ) >. ) , <. (/) , ( _I ` D ) >. ) )
7	3, 5, 6	syl2an 494	. . 3 |- ( ( A = B /\ C = D ) -> rec ( ( a e. om , b e. _V |-> <. suc a , ( a A b ) >. ) , <. (/) , ( _I ` C ) >. ) = rec ( ( a e. om , b e. _V |-> <. suc a , ( a B b ) >. ) , <. (/) , ( _I ` D ) >. ) )
8	7	imaeq1d 5465	. 2 |- ( ( A = B /\ C = D ) -> ( rec ( ( a e. om , b e. _V |-> <. suc a , ( a A b ) >. ) , <. (/) , ( _I ` C ) >. ) " om ) = ( rec ( ( a e. om , b e. _V |-> <. suc a , ( a B b ) >. ) , <. (/) , ( _I ` D ) >. ) " om ) )
9		df-seqom 7543	. 2 |- seq𝜔 ( A , C ) = ( rec ( ( a e. om , b e. _V |-> <. suc a , ( a A b ) >. ) , <. (/) , ( _I ` C ) >. ) " om )
10		df-seqom 7543	. 2 |- seq𝜔 ( B , D ) = ( rec ( ( a e. om , b e. _V |-> <. suc a , ( a B b ) >. ) , <. (/) , ( _I ` D ) >. ) " om )
11	8, 9, 10	3eqtr4g 2681	1 |- ( ( A = B /\ C = D ) -> seq𝜔 ( A , C ) = seq𝜔 ( B , D ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   _Vcvv 3200   (/)c0 3915   <.cop 4183   _I cid 5023   "cima 5117   suc csuc 5725   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   omcom 7065   reccrdg 7505  seq_𝜔cseqom 7542

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-ov 6653  df-oprab 6654  df-mpt2 6655  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543

This theorem is referenced by:  cantnffval  8560  cantnfval  8565  cantnfres  8574  cnfcomlem  8596  cnfcom2  8599  fin23lem33  9167