Theorem ofc12 6153

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 4706	. . . 4 |- ( A X. { B } ) = ( x e. A |-> B )
7	6	a1i 9	. . 3 |- ( ph -> ( A X. { B } ) = ( x e. A |-> B ) )
8		fconstmpt 4706	. . . 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 6146	. 2 |- ( ph -> ( ( A X. { B } ) oF R ( A X. { C } ) ) = ( x e. A |-> ( B R C ) ) )
11		fconstmpt 4706	. 2 |- ( A X. { ( B R C ) } ) = ( x e. A |-> ( B R C ) )
12	10, 11	eqtr4di 2244	1 |- ( ph -> ( ( A X. { B } ) oF R ( A X. { C } ) ) = ( A X. { ( B R C ) } ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   {csn 3618   |-> cmpt 4090   X. cxp 4657  (class class class)co 5918   oFcof 6128

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-of 6130

This theorem is referenced by: (None)