Theorem ofc12 6103

Description: Function operation on two constant functions. (Contributed by Mario Carneiro, 28-Jul-2014.)

Hypotheses

Ref	Expression
ofc12.1	|- ( ph -> A e. V )
ofc12.2	|- ( ph -> B e. W )
ofc12.3	|- ( ph -> C e. X )

Assertion

Ref	Expression
ofc12	|- ( ph -> ( ( A X. { B } ) oF R ( A X. { C } ) ) = ( A X. { ( B R C ) } ) )

Proof of Theorem ofc12

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ofc12.1	. . 3 |- ( ph -> A e. V )
2		ofc12.2	. . . 4 |- ( ph -> B e. W )
3	2	adantr 276	. . 3 |- ( ( ph /\ x e. A ) -> B e. W )
4		ofc12.3	. . . 4 |- ( ph -> C e. X )
5	4	adantr 276	. . 3 |- ( ( ph /\ x e. A ) -> C e. X )
6		fconstmpt 4674	. . . 4 |- ( A X. { B } ) = ( x e. A |-> B )
7	6	a1i 9	. . 3 |- ( ph -> ( A X. { B } ) = ( x e. A |-> B ) )
8		fconstmpt 4674	. . . 4 |- ( A X. { C } ) = ( x e. A |-> C )
9	8	a1i 9	. . 3 |- ( ph -> ( A X. { C } ) = ( x e. A |-> C ) )
10	1, 3, 5, 7, 9	offval2 6098	. 2 |- ( ph -> ( ( A X. { B } ) oF R ( A X. { C } ) ) = ( x e. A |-> ( B R C ) ) )
11		fconstmpt 4674	. 2 |- ( A X. { ( B R C ) } ) = ( x e. A |-> ( B R C ) )
12	10, 11	eqtr4di 2228	1 |- ( ph -> ( ( A X. { B } ) oF R ( A X. { C } ) ) = ( A X. { ( B R C ) } ) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   {csn 3593   |-> cmpt 4065   X. cxp 4625  (class class class)co 5875   oFcof 6081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-setind 4537

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-of 6083

This theorem is referenced by: (None)