Theorem dfco2a 4394

Description: Generalization of dfco2 4393, where C can have any value between dom A i^i ran B and _V. (The proof was 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

Step	Hyp	Ref	Expression
1		ssel 2615	. . . . . . . 8 |- ((dom A i^i ran B) C_ C -> (x e. (dom A i^i ran B) -> x e. C))
2		visset 2295	. . . . . . . . . . . . . . 15 |- x e. _V
3		visset 2295	. . . . . . . . . . . . . . 15 |- w e. _V
4	2, 3	elimasn 4289	. . . . . . . . . . . . . 14 |- (w e. (A"{x}) <-> <.x, w>. e. A)
5	2	opeldm 4160	. . . . . . . . . . . . . 14 |- (<.x, w>. e. A -> x e. dom A)
6	4, 5	sylbi 216	. . . . . . . . . . . . 13 |- (w e. (A"{x}) -> x e. dom A)
7		visset 2295	. . . . . . . . . . . . . . . 16 |- z e. _V
8	7	eliniseg 4294	. . . . . . . . . . . . . . 15 |- (x e. _V -> (z e. (`'B"{x}) <-> zBx))
9	2, 8	ax-mp 7	. . . . . . . . . . . . . 14 |- (z e. (`'B"{x}) <-> zBx)
10	7, 2	brelrn 4191	. . . . . . . . . . . . . 14 |- (zBx -> x e. ran B)
11	9, 10	sylbi 216	. . . . . . . . . . . . 13 |- (z e. (`'B"{x}) -> x e. ran B)
12	6, 11	anim12i 360	. . . . . . . . . . . 12 |- ((w e. (A"{x}) /\ z e. (`'B"{x})) -> (x e. dom A /\ x e. ran B))
13	12	ancoms 484	. . . . . . . . . . 11 |- ((z e. (`'B"{x}) /\ w e. (A"{x})) -> (x e. dom A /\ x e. ran B))
14	13	adantl 424	. . . . . . . . . 10 |- ((y = <.z, w>. /\ (z e. (`'B"{x}) /\ w e. (A"{x}))) -> (x e. dom A /\ x e. ran B))
15	14	19.23aivv 1675	. . . . . . . . 9 |- (E.zE.w(y = <.z, w>. /\ (z e. (`'B"{x}) /\ w e. (A"{x}))) -> (x e. dom A /\ x e. ran B))
16		elxp 4018	. . . . . . . . 9 |- (y e. ((`'B"{x}) X. (A"{x})) <-> E.zE.w(y = <.z, w>. /\ (z e. (`'B"{x}) /\ w e. (A"{x}))))
17		elin 2786	. . . . . . . . 9 |- (x e. (dom A i^i ran B) <-> (x e. dom A /\ x e. ran B))
18	15, 16, 17	3imtr4i 236	. . . . . . . 8 |- (y e. ((`'B"{x}) X. (A"{x})) -> x e. (dom A i^i ran B))
19	1, 18	syl5 20	. . . . . . 7 |- ((dom A i^i ran B) C_ C -> (y e. ((`'B"{x}) X. (A"{x})) -> x e. C))
20	19	pm4.71rd 701	. . . . . 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 1657	. . . . 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 2306	. . . . 5 |- (E.x e. _V y e. ((`'B"{x}) X. (A"{x})) <-> E.x y e. ((`'B"{x}) X. (A"{x})))
23		df-rex 2110	. . . . 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 614	. . . 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 3259	. . . 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 3259	. . . 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 614	. . 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 1882	. 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		dfco2 4393	. 2 |- (A o. B) = U_x e. _V ((`'B"{x}) X. (A"{x}))
30	28, 29	syl5eq 1940	1 |- ((dom A i^i ran B) C_ C -> (A o. B) = U_x e. C ((`'B"{x}) X. (A"{x})))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  E.wrex 2106  _Vcvv 2292   i^i cin 2592   C_ wss 2593  {csn 3044  <.cop 3046  U_ciun 3255   class class class wbr 3338   X. cxp 3984  `'ccnv 3985  dom cdm 3986  ran crn 3987  "cima 3989   o. ccom 3990

This theorem is referenced by:  fparlem3 5083  fparlem4 5084

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-iun 3257  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007

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