Theorem dfco2a 5334

Description: Generalization of dfco2 5333, 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 5333	. 2 |- ( A o. B ) = U_ x e. _V ( ( `' B " { x } ) X. ( A " { x } ) )
2		vex 3047	. . . . . . . . . . . . . 14 |- x e. _V
3		vex 3047	. . . . . . . . . . . . . . 15 |- z e. _V
4	3	eliniseg 5196	. . . . . . . . . . . . . 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 5064	. . . . . . . . . . . . 13 |- ( z B x -> x e. ran B )
7	5, 6	sylbi 199	. . . . . . . . . . . 12 |- ( z e. ( `' B " { x } ) -> x e. ran B )
8		vex 3047	. . . . . . . . . . . . . 14 |- w e. _V
9	2, 8	elimasn 5192	. . . . . . . . . . . . 13 |- ( w e. ( A " { x } ) <-> <. x , w >. e. A )
10	2, 8	opeldm 5037	. . . . . . . . . . . . 13 |- ( <. x , w >. e. A -> x e. dom A )
11	9, 10	sylbi 199	. . . . . . . . . . . 12 |- ( w e. ( A " { x } ) -> x e. dom A )
12	7, 11	anim12ci 570	. . . . . . . . . . 11 |- ( ( z e. ( `' B " { x } ) /\ w e. ( A " { x } ) ) -> ( x e. dom A /\ x e. ran B ) )
13	12	adantl 468	. . . . . . . . . 10 |- ( ( y = <. z , w >. /\ ( z e. ( `' B " { x } ) /\ w e. ( A " { x } ) ) ) -> ( x e. dom A /\ x e. ran B ) )
14	13	exlimivv 1777	. . . . . . . . 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 4850	. . . . . . . . 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 3616	. . . . . . . . 9 |- ( x e. ( dom A i^i ran B ) <-> ( x e. dom A /\ x e. ran B ) )
17	14, 15, 16	3imtr4i 270	. . . . . . . 8 |- ( y e. ( ( `' B " { x } ) X. ( A " { x } ) ) -> x e. ( dom A i^i ran B ) )
18		ssel 3425	. . . . . . . 8 |- ( ( dom A i^i ran B ) C_ C -> ( x e. ( dom A i^i ran B ) -> x e. C ) )
19	17, 18	syl5 33	. . . . . . 7 |- ( ( dom A i^i ran B ) C_ C -> ( y e. ( ( `' B " { x } ) X. ( A " { x } ) ) -> x e. C ) )
20	19	pm4.71rd 640	. . . . . 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 1767	. . . . 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 3061	. . . . 5 |- ( E. x e. _V y e. ( ( `' B " { x } ) X. ( A " { x } ) ) <-> E. x y e. ( ( `' B " { x } ) X. ( A " { x } ) ) )
23		df-rex 2742	. . . . 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 292	. . . 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 4282	. . . 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 4282	. . . 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 292	. . 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 2448	. 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	syl5eq 2496	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   <-> wb 188   /\ wa 371   = wceq 1443   E.wex 1662   e. wcel 1886   E.wrex 2737   _Vcvv 3044   i^i cin 3402   C_ wss 3403   {csn 3967   <.cop 3973   U_ciun 4277   class class class wbr 4401   X. cxp 4831   `'ccnv 4832   dom cdm 4833   ran crn 4834   "cima 4836   o. ccom 4837

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pr 4638

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-iun 4279  df-br 4402  df-opab 4461  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846

This theorem is referenced by:  fparlem3  6895  fparlem4  6896