Theorem dfco2a 5034

Description: Generalization of dfco2 5033, 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 5033	. 2 |- ( A o. B ) = U_ x e. _V ( ( `' B " { x } ) X. ( A " { x } ) )
2		vex 2684	. . . . . . . . . . . . . 14 |- x e. _V
3		vex 2684	. . . . . . . . . . . . . . 15 |- z e. _V
4	3	eliniseg 4904	. . . . . . . . . . . . . 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 4767	. . . . . . . . . . . . 13 |- ( z B x -> x e. ran B )
7	5, 6	sylbi 120	. . . . . . . . . . . 12 |- ( z e. ( `' B " { x } ) -> x e. ran B )
8		vex 2684	. . . . . . . . . . . . . 14 |- w e. _V
9	2, 8	elimasn 4901	. . . . . . . . . . . . 13 |- ( w e. ( A " { x } ) <-> <. x , w >. e. A )
10	2, 8	opeldm 4737	. . . . . . . . . . . . 13 |- ( <. x , w >. e. A -> x e. dom A )
11	9, 10	sylbi 120	. . . . . . . . . . . 12 |- ( w e. ( A " { x } ) -> x e. dom A )
12	7, 11	anim12ci 337	. . . . . . . . . . 11 |- ( ( z e. ( `' B " { x } ) /\ w e. ( A " { x } ) ) -> ( x e. dom A /\ x e. ran B ) )
13	12	adantl 275	. . . . . . . . . 10 |- ( ( y = <. z , w >. /\ ( z e. ( `' B " { x } ) /\ w e. ( A " { x } ) ) ) -> ( x e. dom A /\ x e. ran B ) )
14	13	exlimivv 1868	. . . . . . . . 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 4551	. . . . . . . . 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 3254	. . . . . . . . 9 |- ( x e. ( dom A i^i ran B ) <-> ( x e. dom A /\ x e. ran B ) )
17	14, 15, 16	3imtr4i 200	. . . . . . . 8 |- ( y e. ( ( `' B " { x } ) X. ( A " { x } ) ) -> x e. ( dom A i^i ran B ) )
18		ssel 3086	. . . . . . . 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 391	. . . . . 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 1797	. . . . 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 2699	. . . . 5 |- ( E. x e. _V y e. ( ( `' B " { x } ) X. ( A " { x } ) ) <-> E. x y e. ( ( `' B " { x } ) X. ( A " { x } ) ) )
23		df-rex 2420	. . . . 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 222	. . . 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 3812	. . . 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 3812	. . . 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 222	. . 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 2135	. 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 2182	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 103   <-> wb 104   = wceq 1331   E.wex 1468   e. wcel 1480   E.wrex 2415   _Vcvv 2681   i^i cin 3065   C_ wss 3066   {csn 3522   <.cop 3525   U_ciun 3808   class class class wbr 3924   X. cxp 4532   `'ccnv 4533   dom cdm 4534   ran crn 4535   "cima 4537   o. ccom 4538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-iun 3810  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547

This theorem is referenced by: (None)