Theorem dfco2a 5170

Description: Generalization of dfco2 5169, where C can have any value between dom A i^i ran B and _V. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
dfco2a	|- ( ( dom A i^i ran B ) C_ C -> ( A o. B ) = U_ x e. C ( ( `' B " { x } ) X. ( A " { x } ) ) )

Distinct variable groups:   x, A   x, B   x, C

Proof of Theorem dfco2a

Dummy variables w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfco2 5169	. 2 |- ( A o. B ) = U_ x e. _V ( ( `' B " { x } ) X. ( A " { x } ) )
2		vex 2766	. . . . . . . . . . . . . 14 |- x e. _V
3		vex 2766	. . . . . . . . . . . . . . 15 |- z e. _V
4	3	eliniseg 5039	. . . . . . . . . . . . . 14 |- ( x e. _V -> ( z e. ( `' B " { x } ) <-> z B x ) )
5	2, 4	ax-mp 5	. . . . . . . . . . . . 13 |- ( z e. ( `' B " { x } ) <-> z B x )
6	3, 2	brelrn 4899	. . . . . . . . . . . . 13 |- ( z B x -> x e. ran B )
7	5, 6	sylbi 121	. . . . . . . . . . . 12 |- ( z e. ( `' B " { x } ) -> x e. ran B )
8		vex 2766	. . . . . . . . . . . . . 14 |- w e. _V
9	2, 8	elimasn 5036	. . . . . . . . . . . . 13 |- ( w e. ( A " { x } ) <-> <. x , w >. e. A )
10	2, 8	opeldm 4869	. . . . . . . . . . . . 13 |- ( <. x , w >. e. A -> x e. dom A )
11	9, 10	sylbi 121	. . . . . . . . . . . 12 |- ( w e. ( A " { x } ) -> x e. dom A )
12	7, 11	anim12ci 339	. . . . . . . . . . 11 |- ( ( z e. ( `' B " { x } ) /\ w e. ( A " { x } ) ) -> ( x e. dom A /\ x e. ran B ) )
13	12	adantl 277	. . . . . . . . . 10 |- ( ( y = <. z , w >. /\ ( z e. ( `' B " { x } ) /\ w e. ( A " { x } ) ) ) -> ( x e. dom A /\ x e. ran B ) )
14	13	exlimivv 1911	. . . . . . . . 9 |- ( E. z E. w ( y = <. z , w >. /\ ( z e. ( `' B " { x } ) /\ w e. ( A " { x } ) ) ) -> ( x e. dom A /\ x e. ran B ) )
15		elxp 4680	. . . . . . . . 9 |- ( y e. ( ( `' B " { x } ) X. ( A " { x } ) ) <-> E. z E. w ( y = <. z , w >. /\ ( z e. ( `' B " { x } ) /\ w e. ( A " { x } ) ) ) )
16		elin 3346	. . . . . . . . 9 |- ( x e. ( dom A i^i ran B ) <-> ( x e. dom A /\ x e. ran B ) )
17	14, 15, 16	3imtr4i 201	. . . . . . . 8 |- ( y e. ( ( `' B " { x } ) X. ( A " { x } ) ) -> x e. ( dom A i^i ran B ) )
18		ssel 3177	. . . . . . . 8 |- ( ( dom A i^i ran B ) C_ C -> ( x e. ( dom A i^i ran B ) -> x e. C ) )
19	17, 18	syl5 32	. . . . . . 7 |- ( ( dom A i^i ran B ) C_ C -> ( y e. ( ( `' B " { x } ) X. ( A " { x } ) ) -> x e. C ) )
20	19	pm4.71rd 394	. . . . . 6 |- ( ( dom A i^i ran B ) C_ C -> ( y e. ( ( `' B " { x } ) X. ( A " { x } ) ) <-> ( x e. C /\ y e. ( ( `' B " { x } ) X. ( A " { x } ) ) ) ) )
21	20	exbidv 1839	. . . . 5 |- ( ( dom A i^i ran B ) C_ C -> ( E. x y e. ( ( `' B " { x } ) X. ( A " { x } ) ) <-> E. x ( x e. C /\ y e. ( ( `' B " { x } ) X. ( A " { x } ) ) ) ) )
22		rexv 2781	. . . . 5 |- ( E. x e. _V y e. ( ( `' B " { x } ) X. ( A " { x } ) ) <-> E. x y e. ( ( `' B " { x } ) X. ( A " { x } ) ) )
23		df-rex 2481	. . . . 5 |- ( E. x e. C y e. ( ( `' B " { x } ) X. ( A " { x } ) ) <-> E. x ( x e. C /\ y e. ( ( `' B " { x } ) X. ( A " { x } ) ) ) )
24	21, 22, 23	3bitr4g 223	. . . 4 |- ( ( dom A i^i ran B ) C_ C -> ( E. x e. _V y e. ( ( `' B " { x } ) X. ( A " { x } ) ) <-> E. x e. C y e. ( ( `' B " { x } ) X. ( A " { x } ) ) ) )
25		eliun 3920	. . . 4 |- ( y e. U_ x e. _V ( ( `' B " { x } ) X. ( A " { x } ) ) <-> E. x e. _V y e. ( ( `' B " { x } ) X. ( A " { x } ) ) )
26		eliun 3920	. . . 4 |- ( y e. U_ x e. C ( ( `' B " { x } ) X. ( A " { x } ) ) <-> E. x e. C y e. ( ( `' B " { x } ) X. ( A " { x } ) ) )
27	24, 25, 26	3bitr4g 223	. . 3 |- ( ( dom A i^i ran B ) C_ C -> ( y e. U_ x e. _V ( ( `' B " { x } ) X. ( A " { x } ) ) <-> y e. U_ x e. C ( ( `' B " { x } ) X. ( A " { x } ) ) ) )
28	27	eqrdv 2194	. 2 |- ( ( dom A i^i ran B ) C_ C -> U_ x e. _V ( ( `' B " { x } ) X. ( A " { x } ) ) = U_ x e. C ( ( `' B " { x } ) X. ( A " { x } ) ) )
29	1, 28	eqtrid 2241	1 |- ( ( dom A i^i ran B ) C_ C -> ( A o. B ) = U_ x e. C ( ( `' B " { x } ) X. ( A " { x } ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   E.wex 1506   e. wcel 2167   E.wrex 2476   _Vcvv 2763   i^i cin 3156   C_ wss 3157   {csn 3622   <.cop 3625   U_ciun 3916   class class class wbr 4033   X. cxp 4661   `'ccnv 4662   dom cdm 4663   ran crn 4664   "cima 4666   o. ccom 4667

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-iun 3918  df-br 4034  df-opab 4095  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676

This theorem is referenced by: (None)