Theorem dfco2a 4764

Description: Generalization of dfco2 4763, 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 4763	. 2 |- (A o. B) = U_x e. _V (( `'B " {x }) X. (A " {x }))
2		vex 2554	. . . . . . . . . . . . . 14 |- x e. _V
3		vex 2554	. . . . . . . . . . . . . . 15 |- z e. _V
4	3	eliniseg 4638	. . . . . . . . . . . . . 14 |- (x e. _V -> (z e. ( `'B " {x }) <-> zBx))
5	2, 4	ax-mp 7	. . . . . . . . . . . . 13 |- (z e. ( `'B " {x }) <-> zBx)
6	3, 2	brelrn 4510	. . . . . . . . . . . . 13 |- (zBx -> x e. ran B)
7	5, 6	sylbi 114	. . . . . . . . . . . 12 |- (z e. ( `'B " {x }) -> x e. ran B)
8		vex 2554	. . . . . . . . . . . . . 14 |- w e. _V
9	2, 8	elimasn 4635	. . . . . . . . . . . . 13 |- (w e. (A " {x }) <-> <.x , w >. e. A)
10	2, 8	opeldm 4481	. . . . . . . . . . . . 13 |- ( <.x , w >. e. A -> x e. dom A)
11	9, 10	sylbi 114	. . . . . . . . . . . 12 |- (w e. (A " {x }) -> x e. dom A)
12	7, 11	anim12ci 322	. . . . . . . . . . 11 |- ((z e. ( `'B " {x }) /\ w e. (A " {x })) -> (x e. dom A /\ x e. ran B))
13	12	adantl 262	. . . . . . . . . 10 |- ((y = <.z , w >. /\ (z e. ( `'B " {x }) /\ w e. (A " {x }))) -> (x e. dom A /\ x e. ran B))
14	13	exlimivv 1773	. . . . . . . . 9 |- (E.zE.w(y = <.z , w >. /\ (z e. ( `'B " {x }) /\ w e. (A " {x }))) -> (x e. dom A /\ x e. ran B))
15		elxp 4305	. . . . . . . . 9 |- (y e. (( `'B " {x }) X. (A " {x })) <-> E.zE.w(y = <.z , w >. /\ (z e. ( `'B " {x }) /\ w e. (A " {x }))))
16		elin 3120	. . . . . . . . 9 |- (x e. ( dom A i^i ran B) <-> (x e. dom A /\ x e. ran B))
17	14, 15, 16	3imtr4i 190	. . . . . . . 8 |- (y e. (( `'B " {x }) X. (A " {x })) -> x e. ( dom A i^i ran B))
18		ssel 2933	. . . . . . . 8 |- (( dom A i^i ran B) C_ C -> (x e. ( dom A i^i ran B) -> x e. C))
19	17, 18	syl5 28	. . . . . . 7 |- (( dom A i^i ran B) C_ C -> (y e. (( `'B " {x }) X. (A " {x })) -> x e. C))
20	19	pm4.71rd 374	. . . . . 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 1703	. . . . 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 2566	. . . . 5 |- (E.x e. _V y e. (( `'B " {x }) X. (A " {x })) <-> E.x y e. (( `'B " {x }) X. (A " {x })))
23		df-rex 2306	. . . . 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 212	. . . 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 3652	. . . 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 3652	. . . 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 212	. . 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 2035	. 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 2081	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 97   <-> wb 98   = wceq 1242  E.wex 1378   e. wcel 1390  E.wrex 2301   _Vcvv 2551   i^i cin 2910   C_ wss 2911   {csn 3367   <.cop 3370   U_ciun 3648   class class class wbr 3755   X. cxp 4286   `'ccnv 4287   dom cdm 4288   ran crn 4289   "cima 4291   o. ccom 4292

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-iun 3650  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301

This theorem is referenced by: (None)