Theorem clintopval 41840

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

Assertion

Ref	Expression
clintopval	|- ( M e. V -> ( clIntOp ` M ) = ( M ^m ( M X. M ) ) )

Proof of Theorem clintopval

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-clintop 41836	. . 3 |- clIntOp = ( m e. _V |-> ( m intOp m ) )
2	1	a1i 11	. 2 |- ( M e. V -> clIntOp = ( m e. _V |-> ( m intOp m ) ) )
3		id 22	. . . 4 |- ( m = M -> m = M )
4	3, 3	oveq12d 6668	. . 3 |- ( m = M -> ( m intOp m ) = ( M intOp M ) )
5		intopval 41838	. . . 4 |- ( ( M e. V /\ M e. V ) -> ( M intOp M ) = ( M ^m ( M X. M ) ) )
6	5	anidms 677	. . 3 |- ( M e. V -> ( M intOp M ) = ( M ^m ( M X. M ) ) )
7	4, 6	sylan9eqr 2678	. 2 |- ( ( M e. V /\ m = M ) -> ( m intOp m ) = ( M ^m ( M X. M ) ) )
8		elex 3212	. 2 |- ( M e. V -> M e. _V )
9		ovexd 6680	. 2 |- ( M e. V -> ( M ^m ( M X. M ) ) e. _V )
10	2, 7, 8, 9	fvmptd 6288	1 |- ( M e. V -> ( clIntOp ` M ) = ( M ^m ( M X. M ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   |-> cmpt 4729   X. cxp 5112   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   intOp cintop 41832   clIntOp cclintop 41833

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-csb 3534  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-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  df-clintop 41836

This theorem is referenced by:  assintopmap  41842  isclintop  41843