Theorem inimasn 5047

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 3319	. . 3 |- ( x e. ( ( A " { C } ) i^i ( B " { C } ) ) <-> ( x e. ( A " { C } ) /\ x e. ( B " { C } ) ) )
2		elin 3319	. . . . 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 2741	. . . . 5 |- x e. _V
5		elimasng 4997	. . . . 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 4997	. . . . . 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 4997	. . . . . 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 2175	1 |- ( C e. V -> ( ( A i^i B ) " { C } ) = ( ( A " { C } ) i^i ( B " { C } ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   _Vcvv 2738   i^i cin 3129   {csn 3593   <.cop 3596   "cima 4630

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 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-sep 4122  ax-pow 4175  ax-pr 4210

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  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-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-opab 4066  df-xp 4633  df-cnv 4635  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640

This theorem is referenced by: (None)