Theorem inimasn 5550

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 3796	. . 3 |- ( x e. ( ( A " { C } ) i^i ( B " { C } ) ) <-> ( x e. ( A " { C } ) /\ x e. ( B " { C } ) ) )
2		elin 3796	. . . . 5 |- ( <. C , x >. e. ( A i^i B ) <-> ( <. C , x >. e. A /\ <. C , x >. e. B ) )
3	2	a1i 11	. . . 4 |- ( C e. V -> ( <. C , x >. e. ( A i^i B ) <-> ( <. C , x >. e. A /\ <. C , x >. e. B ) ) )
4		vex 3203	. . . . 5 |- x e. _V
5		elimasng 5491	. . . . 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 707	. . . 4 |- ( C e. V -> ( x e. ( ( A i^i B ) " { C } ) <-> <. C , x >. e. ( A i^i B ) ) )
7		elimasng 5491	. . . . . 6 |- ( ( C e. V /\ x e. _V ) -> ( x e. ( A " { C } ) <-> <. C , x >. e. A ) )
8	4, 7	mpan2 707	. . . . 5 |- ( C e. V -> ( x e. ( A " { C } ) <-> <. C , x >. e. A ) )
9		elimasng 5491	. . . . . 6 |- ( ( C e. V /\ x e. _V ) -> ( x e. ( B " { C } ) <-> <. C , x >. e. B ) )
10	4, 9	mpan2 707	. . . . 5 |- ( C e. V -> ( x e. ( B " { C } ) <-> <. C , x >. e. B ) )
11	8, 10	anbi12d 747	. . . 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 301	. . 3 |- ( C e. V -> ( ( x e. ( A " { C } ) /\ x e. ( B " { C } ) ) <-> x e. ( ( A i^i B ) " { C } ) ) )
13	1, 12	syl5rbb 273	. 2 |- ( C e. V -> ( x e. ( ( A i^i B ) " { C } ) <-> x e. ( ( A " { C } ) i^i ( B " { C } ) ) ) )
14	13	eqrdv 2620	1 |- ( C e. V -> ( ( A i^i B ) " { C } ) = ( ( A " { C } ) i^i ( B " { C } ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   {csn 4177   <.cop 4183   "cima 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  restutopopn  22042  ustuqtop2  22046