Theorem intopval 41838

Description: The internal (binary) operations for a set. (Contributed by AV, 20-Jan-2020.)

Assertion

Ref	Expression
intopval	|- ( ( M e. V /\ N e. W ) -> ( M intOp N ) = ( N ^m ( M X. M ) ) )

Proof of Theorem intopval

Dummy variables m n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-intop 41835	. . 3 |- intOp = ( m e. _V , n e. _V |-> ( n ^m ( m X. m ) ) )
2	1	a1i 11	. 2 |- ( ( M e. V /\ N e. W ) -> intOp = ( m e. _V , n e. _V |-> ( n ^m ( m X. m ) ) ) )
3		simpr 477	. . . 4 |- ( ( m = M /\ n = N ) -> n = N )
4		simpl 473	. . . . 5 |- ( ( m = M /\ n = N ) -> m = M )
5	4	sqxpeqd 5141	. . . 4 |- ( ( m = M /\ n = N ) -> ( m X. m ) = ( M X. M ) )
6	3, 5	oveq12d 6668	. . 3 |- ( ( m = M /\ n = N ) -> ( n ^m ( m X. m ) ) = ( N ^m ( M X. M ) ) )
7	6	adantl 482	. 2 |- ( ( ( M e. V /\ N e. W ) /\ ( m = M /\ n = N ) ) -> ( n ^m ( m X. m ) ) = ( N ^m ( M X. M ) ) )
8		elex 3212	. . 3 |- ( M e. V -> M e. _V )
9	8	adantr 481	. 2 |- ( ( M e. V /\ N e. W ) -> M e. _V )
10		elex 3212	. . 3 |- ( N e. W -> N e. _V )
11	10	adantl 482	. 2 |- ( ( M e. V /\ N e. W ) -> N e. _V )
12		ovexd 6680	. 2 |- ( ( M e. V /\ N e. W ) -> ( N ^m ( M X. M ) ) e. _V )
13	2, 7, 9, 11, 12	ovmpt2d 6788	1 |- ( ( M e. V /\ N e. W ) -> ( M intOp N ) = ( N ^m ( M X. M ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   X. cxp 5112  (class class class)co 6650   |-> cmpt2 6652   ^m cmap 7857   intOp cintop 41832

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-eu 2474  df-mo 2475  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-sbc 3436  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-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-intop 41835

This theorem is referenced by:  intop  41839  clintopval  41840