Theorem ofc12 6261

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 4772	. . . 4 |- ( A X. { B } ) = ( x e. A |-> B )
7	6	a1i 9	. . 3 |- ( ph -> ( A X. { B } ) = ( x e. A |-> B ) )
8		fconstmpt 4772	. . . 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 6253	. 2 |- ( ph -> ( ( A X. { B } ) oF R ( A X. { C } ) ) = ( x e. A |-> ( B R C ) ) )
11		fconstmpt 4772	. 2 |- ( A X. { ( B R C ) } ) = ( x e. A |-> ( B R C ) )
12	10, 11	eqtr4di 2281	1 |- ( ph -> ( ( A X. { B } ) oF R ( A X. { C } ) ) = ( A X. { ( B R C ) } ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2201   {csn 3668   |-> cmpt 4149   X. cxp 4722  (class class class)co 6020   oFcof 6235

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-coll 4203  ax-sep 4206  ax-pow 4263  ax-pr 4298  ax-setind 4634

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-pw 3653  df-sn 3674  df-pr 3675  df-op 3677  df-uni 3893  df-iun 3971  df-br 4088  df-opab 4150  df-mpt 4151  df-id 4389  df-xp 4730  df-rel 4731  df-cnv 4732  df-co 4733  df-dm 4734  df-rn 4735  df-res 4736  df-ima 4737  df-iota 5285  df-fun 5327  df-fn 5328  df-f 5329  df-f1 5330  df-fo 5331  df-f1o 5332  df-fv 5333  df-ov 6023  df-oprab 6024  df-mpo 6025  df-of 6237

This theorem is referenced by: (None)