Theorem inimasn 5000

Description: The intersection of the image of singleton (Contributed by Thierry Arnoux, 16-Dec-2017.)

Assertion

Ref	Expression
inimasn	|- ( C e. V -> ( ( A i^i B ) " { C } ) = ( ( A " { C } ) i^i ( B " { C } ) ) )

Proof of Theorem inimasn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3290	. . 3 |- ( x e. ( ( A " { C } ) i^i ( B " { C } ) ) <-> ( x e. ( A " { C } ) /\ x e. ( B " { C } ) ) )
2		elin 3290	. . . . 5 |- ( <. C , x >. e. ( A i^i B ) <-> ( <. C , x >. e. A /\ <. C , x >. e. B ) )
3	2	a1i 9	. . . 4 |- ( C e. V -> ( <. C , x >. e. ( A i^i B ) <-> ( <. C , x >. e. A /\ <. C , x >. e. B ) ) )
4		vex 2715	. . . . 5 |- x e. _V
5		elimasng 4951	. . . . 5 |- ( ( C e. V /\ x e. _V ) -> ( x e. ( ( A i^i B ) " { C } ) <-> <. C , x >. e. ( A i^i B ) ) )
6	4, 5	mpan2 422	. . . 4 |- ( C e. V -> ( x e. ( ( A i^i B ) " { C } ) <-> <. C , x >. e. ( A i^i B ) ) )
7		elimasng 4951	. . . . . 6 |- ( ( C e. V /\ x e. _V ) -> ( x e. ( A " { C } ) <-> <. C , x >. e. A ) )
8	4, 7	mpan2 422	. . . . 5 |- ( C e. V -> ( x e. ( A " { C } ) <-> <. C , x >. e. A ) )
9		elimasng 4951	. . . . . 6 |- ( ( C e. V /\ x e. _V ) -> ( x e. ( B " { C } ) <-> <. C , x >. e. B ) )
10	4, 9	mpan2 422	. . . . 5 |- ( C e. V -> ( x e. ( B " { C } ) <-> <. C , x >. e. B ) )
11	8, 10	anbi12d 465	. . . 4 |- ( C e. V -> ( ( x e. ( A " { C } ) /\ x e. ( B " { C } ) ) <-> ( <. C , x >. e. A /\ <. C , x >. e. B ) ) )
12	3, 6, 11	3bitr4rd 220	. . 3 |- ( C e. V -> ( ( x e. ( A " { C } ) /\ x e. ( B " { C } ) ) <-> x e. ( ( A i^i B ) " { C } ) ) )
13	1, 12	syl5rbb 192	. 2 |- ( C e. V -> ( x e. ( ( A i^i B ) " { C } ) <-> x e. ( ( A " { C } ) i^i ( B " { C } ) ) ) )
14	13	eqrdv 2155	1 |- ( C e. V -> ( ( A i^i B ) " { C } ) = ( ( A " { C } ) i^i ( B " { C } ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   e. wcel 2128   _Vcvv 2712   i^i cin 3101   {csn 3560   <.cop 3563   "cima 4586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-xp 4589  df-cnv 4591  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596

This theorem is referenced by: (None)