Theorem inimasn 5161

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 3392	. . 3 |- ( x e. ( ( A " { C } ) i^i ( B " { C } ) ) <-> ( x e. ( A " { C } ) /\ x e. ( B " { C } ) ) )
2		elin 3392	. . . . 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 2806	. . . . 5 |- x e. _V
5		elimasng 5111	. . . . 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 425	. . . 4 |- ( C e. V -> ( x e. ( ( A i^i B ) " { C } ) <-> <. C , x >. e. ( A i^i B ) ) )
7		elimasng 5111	. . . . . 6 |- ( ( C e. V /\ x e. _V ) -> ( x e. ( A " { C } ) <-> <. C , x >. e. A ) )
8	4, 7	mpan2 425	. . . . 5 |- ( C e. V -> ( x e. ( A " { C } ) <-> <. C , x >. e. A ) )
9		elimasng 5111	. . . . . 6 |- ( ( C e. V /\ x e. _V ) -> ( x e. ( B " { C } ) <-> <. C , x >. e. B ) )
10	4, 9	mpan2 425	. . . . 5 |- ( C e. V -> ( x e. ( B " { C } ) <-> <. C , x >. e. B ) )
11	8, 10	anbi12d 473	. . . 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 221	. . 3 |- ( C e. V -> ( ( x e. ( A " { C } ) /\ x e. ( B " { C } ) ) <-> x e. ( ( A i^i B ) " { C } ) ) )
13	1, 12	bitr2id 193	. 2 |- ( C e. V -> ( x e. ( ( A i^i B ) " { C } ) <-> x e. ( ( A " { C } ) i^i ( B " { C } ) ) ) )
14	13	eqrdv 2229	1 |- ( C e. V -> ( ( A i^i B ) " { C } ) = ( ( A " { C } ) i^i ( B " { C } ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   _Vcvv 2803   i^i cin 3200   {csn 3673   <.cop 3676   "cima 4734

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-cnv 4739  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744

This theorem is referenced by: (None)