Theorem inimasn 4761

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 3155	. . 3 |- ( x e. ( ( A " { C } ) i^i ( B " { C } ) ) <-> ( x e. ( A " { C } ) /\ x e. ( B " { C } ) ) )
2		elin 3155	. . . . 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 2604	. . . . 5 |- x e. _V
5		elimasng 4713	. . . . 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 415	. . . 4 |- ( C e. V -> ( x e. ( ( A i^i B ) " { C } ) <-> <. C , x >. e. ( A i^i B ) ) )
7		elimasng 4713	. . . . . 6 |- ( ( C e. V /\ x e. _V ) -> ( x e. ( A " { C } ) <-> <. C , x >. e. A ) )
8	4, 7	mpan2 415	. . . . 5 |- ( C e. V -> ( x e. ( A " { C } ) <-> <. C , x >. e. A ) )
9		elimasng 4713	. . . . . 6 |- ( ( C e. V /\ x e. _V ) -> ( x e. ( B " { C } ) <-> <. C , x >. e. B ) )
10	4, 9	mpan2 415	. . . . 5 |- ( C e. V -> ( x e. ( B " { C } ) <-> <. C , x >. e. B ) )
11	8, 10	anbi12d 456	. . . 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 219	. . 3 |- ( C e. V -> ( ( x e. ( A " { C } ) /\ x e. ( B " { C } ) ) <-> x e. ( ( A i^i B ) " { C } ) ) )
13	1, 12	syl5rbb 191	. 2 |- ( C e. V -> ( x e. ( ( A i^i B ) " { C } ) <-> x e. ( ( A " { C } ) i^i ( B " { C } ) ) ) )
14	13	eqrdv 2079	1 |- ( C e. V -> ( ( A i^i B ) " { C } ) = ( ( A " { C } ) i^i ( B " { C } ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   _Vcvv 2601   i^i cin 2972   {csn 3398   <.cop 3401   "cima 4366

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376

This theorem is referenced by: (None)