Theorem finxpeq1 33223

Description: Equality theorem for Cartesian exponentiation. (Contributed by ML, 19-Oct-2020.)

Assertion

Ref	Expression
finxpeq1	|- ( U = V -> ( U ^^ ^^ N ) = ( V ^^ ^^ N ) )

Proof of Theorem finxpeq1

Dummy variables n x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2690	. . . . . . . . . 10 |- ( U = V -> ( x e. U <-> x e. V ) )
2	1	anbi2d 740	. . . . . . . . 9 |- ( U = V -> ( ( n = 1o /\ x e. U ) <-> ( n = 1o /\ x e. V ) ) )
3		xpeq2 5129	. . . . . . . . . . 11 |- ( U = V -> ( _V X. U ) = ( _V X. V ) )
4	3	eleq2d 2687	. . . . . . . . . 10 |- ( U = V -> ( x e. ( _V X. U ) <-> x e. ( _V X. V ) ) )
5	4	ifbid 4108	. . . . . . . . 9 |- ( U = V -> if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) = if ( x e. ( _V X. V ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) )
6	2, 5	ifbieq2d 4111	. . . . . . . 8 |- ( U = V -> if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) = if ( ( n = 1o /\ x e. V ) , (/) , if ( x e. ( _V X. V ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) )
7	6	mpt2eq3dv 6721	. . . . . . 7 |- ( U = V -> ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) ) = ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. V ) , (/) , if ( x e. ( _V X. V ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) ) )
8		rdgeq1 7507	. . . . . . 7 |- ( ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) ) = ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. V ) , (/) , if ( x e. ( _V X. V ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) ) -> rec ( ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) ) , <. N , y >. ) = rec ( ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. V ) , (/) , if ( x e. ( _V X. V ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) ) , <. N , y >. ) )
9	7, 8	syl 17	. . . . . 6 |- ( U = V -> rec ( ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) ) , <. N , y >. ) = rec ( ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. V ) , (/) , if ( x e. ( _V X. V ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) ) , <. N , y >. ) )
10	9	fveq1d 6193	. . . . 5 |- ( U = V -> ( rec ( ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) ) , <. N , y >. ) ` N ) = ( rec ( ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. V ) , (/) , if ( x e. ( _V X. V ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) ) , <. N , y >. ) ` N ) )
11	10	eqeq2d 2632	. . . 4 |- ( U = V -> ( (/) = ( rec ( ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) ) , <. N , y >. ) ` N ) <-> (/) = ( rec ( ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. V ) , (/) , if ( x e. ( _V X. V ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) ) , <. N , y >. ) ` N ) ) )
12	11	anbi2d 740	. . 3 |- ( U = V -> ( ( N e. om /\ (/) = ( rec ( ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) ) , <. N , y >. ) ` N ) ) <-> ( N e. om /\ (/) = ( rec ( ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. V ) , (/) , if ( x e. ( _V X. V ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) ) , <. N , y >. ) ` N ) ) ) )
13	12	abbidv 2741	. 2 |- ( U = V -> { y | ( N e. om /\ (/) = ( rec ( ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) ) , <. N , y >. ) ` N ) ) } = { y | ( N e. om /\ (/) = ( rec ( ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. V ) , (/) , if ( x e. ( _V X. V ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) ) , <. N , y >. ) ` N ) ) } )
14		df-finxp 33221	. 2 |- ( U ^^ ^^ N ) = { y | ( N e. om /\ (/) = ( rec ( ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) ) , <. N , y >. ) ` N ) ) }
15		df-finxp 33221	. 2 |- ( V ^^ ^^ N ) = { y | ( N e. om /\ (/) = ( rec ( ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. V ) , (/) , if ( x e. ( _V X. V ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) ) , <. N , y >. ) ` N ) ) }
16	13, 14, 15	3eqtr4g 2681	1 |- ( U = V -> ( U ^^ ^^ N ) = ( V ^^ ^^ N ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   (/)c0 3915   ifcif 4086   <.cop 4183   U.cuni 4436   X. cxp 5112   ` cfv 5888   |-> cmpt2 6652   omcom 7065   1stc1st 7166   reccrdg 7505   1oc1o 7553   ^^ ^^cfinxp 33220

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-oprab 6654  df-mpt2 6655  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-finxp 33221

This theorem is referenced by: (None)