Theorem dfco2aOLD 4395

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

Assertion

Ref	Expression
dfco2aOLD	|- ((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 dfco2aOLD

Step	Hyp	Ref	Expression
1		iunss1 3266	. . 3 |- ((dom A i^i ran B) C_ C -> U_x e. (dom A i^i ran B)((`'B"{x}) X. (A"{x})) C_ U_x e. C ((`'B"{x}) X. (A"{x})))
2		dfco2 4393	. . . 4 |- (A o. B) = U_x e. _V ((`'B"{x}) X. (A"{x}))
3		undifv 2948	. . . . 5 |- ((dom A i^i ran B) u. (_V \ (dom A i^i ran B))) = _V
4		iuneq1 3269	. . . . 5 |- (((dom A i^i ran B) u. (_V \ (dom A i^i ran B))) = _V -> U_x e. ((dom A i^i ran B) u. (_V \ (dom A i^i ran B)))((`'B"{x}) X. (A"{x})) = U_x e. _V ((`'B"{x}) X. (A"{x})))
5	3, 4	ax-mp 7	. . . 4 |- U_x e. ((dom A i^i ran B) u. (_V \ (dom A i^i ran B)))((`'B"{x}) X. (A"{x})) = U_x e. _V ((`'B"{x}) X. (A"{x}))
6		iunxun 3329	. . . . 5 |- U_x e. ((dom A i^i ran B) u. (_V \ (dom A i^i ran B)))((`'B"{x}) X. (A"{x})) = (U_x e. (dom A i^i ran B)((`'B"{x}) X. (A"{x})) u. U_x e. (_V \ (dom A i^i ran B))((`'B"{x}) X. (A"{x})))
7		df-rn 4005	. . . . . . . . . . . . 13 |- ran B = dom `' B
8	7	eleq2i 1961	. . . . . . . . . . . 12 |- (x e. ran B <-> x e. dom `' B)
9	8	notbii 204	. . . . . . . . . . 11 |- (-. x e. ran B <-> -. x e. dom `' B)
10		ndmima 4300	. . . . . . . . . . 11 |- (-. x e. dom `' B -> (`'B"{x}) = (/))
11	9, 10	sylbi 216	. . . . . . . . . 10 |- (-. x e. ran B -> (`'B"{x}) = (/))
12		ndmima 4300	. . . . . . . . . 10 |- (-. x e. dom A -> (A"{x}) = (/))
13	11, 12	orim12i 363	. . . . . . . . 9 |- ((-. x e. ran B \/ -. x e. dom A) -> ((`'B"{x}) = (/) \/ (A"{x}) = (/)))
14		eldif 2609	. . . . . . . . . 10 |- (x e. (_V \ (dom A i^i ran B)) <-> (x e. _V /\ -. x e. (dom A i^i ran B)))
15		visset 2295	. . . . . . . . . . 11 |- x e. _V
16	15	biantrur 794	. . . . . . . . . 10 |- (-. x e. (dom A i^i ran B) <-> (x e. _V /\ -. x e. (dom A i^i ran B)))
17		elin 2786	. . . . . . . . . . . 12 |- (x e. (dom A i^i ran B) <-> (x e. dom A /\ x e. ran B))
18	17	notbii 204	. . . . . . . . . . 11 |- (-. x e. (dom A i^i ran B) <-> -. (x e. dom A /\ x e. ran B))
19		ianor 329	. . . . . . . . . . 11 |- (-. (x e. dom A /\ x e. ran B) <-> (-. x e. dom A \/ -. x e. ran B))
20		orcom 266	. . . . . . . . . . 11 |- ((-. x e. dom A \/ -. x e. ran B) <-> (-. x e. ran B \/ -. x e. dom A))
21	18, 19, 20	3bitri 194	. . . . . . . . . 10 |- (-. x e. (dom A i^i ran B) <-> (-. x e. ran B \/ -. x e. dom A))
22	14, 16, 21	3bitr2i 196	. . . . . . . . 9 |- (x e. (_V \ (dom A i^i ran B)) <-> (-. x e. ran B \/ -. x e. dom A))
23		xpeq0 4336	. . . . . . . . 9 |- (((`'B"{x}) X. (A"{x})) = (/) <-> ((`'B"{x}) = (/) \/ (A"{x}) = (/)))
24	13, 22, 23	3imtr4i 236	. . . . . . . 8 |- (x e. (_V \ (dom A i^i ran B)) -> ((`'B"{x}) X. (A"{x})) = (/))
25	24	iuneq2i 3276	. . . . . . 7 |- U_x e. (_V \ (dom A i^i ran B))((`'B"{x}) X. (A"{x})) = U_x e. (_V \ (dom A i^i ran B))(/)
26		iun0 3309	. . . . . . 7 |- U_x e. (_V \ (dom A i^i ran B))(/) = (/)
27	25, 26	eqtri 1908	. . . . . 6 |- U_x e. (_V \ (dom A i^i ran B))((`'B"{x}) X. (A"{x})) = (/)
28	27	uneq2i 2752	. . . . 5 |- (U_x e. (dom A i^i ran B)((`'B"{x}) X. (A"{x})) u. U_x e. (_V \ (dom A i^i ran B))((`'B"{x}) X. (A"{x}))) = (U_x e. (dom A i^i ran B)((`'B"{x}) X. (A"{x})) u. (/))
29		un0 2896	. . . . 5 |- (U_x e. (dom A i^i ran B)((`'B"{x}) X. (A"{x})) u. (/)) = U_x e. (dom A i^i ran B)((`'B"{x}) X. (A"{x}))
30	6, 28, 29	3eqtri 1912	. . . 4 |- U_x e. ((dom A i^i ran B) u. (_V \ (dom A i^i ran B)))((`'B"{x}) X. (A"{x})) = U_x e. (dom A i^i ran B)((`'B"{x}) X. (A"{x}))
31	2, 5, 30	3eqtr2i 1915	. . 3 |- (A o. B) = U_x e. (dom A i^i ran B)((`'B"{x}) X. (A"{x}))
32	1, 31	syl5ss 2661	. 2 |- ((dom A i^i ran B) C_ C -> (A o. B) C_ U_x e. C ((`'B"{x}) X. (A"{x})))
33		ssv 2636	. . . . 5 |- C C_ _V
34		iunss1 3266	. . . . 5 |- (C C_ _V -> U_x e. C ((`'B"{x}) X. (A"{x})) C_ U_x e. _V ((`'B"{x}) X. (A"{x})))
35	33, 34	ax-mp 7	. . . 4 |- U_x e. C ((`'B"{x}) X. (A"{x})) C_ U_x e. _V ((`'B"{x}) X. (A"{x}))
36	35	a1i 8	. . 3 |- ((dom A i^i ran B) C_ C -> U_x e. C ((`'B"{x}) X. (A"{x})) C_ U_x e. _V ((`'B"{x}) X. (A"{x})))
37	36, 2	syl6ssr 2664	. 2 |- ((dom A i^i ran B) C_ C -> U_x e. C ((`'B"{x}) X. (A"{x})) C_ (A o. B))
38	32, 37	eqssd 2633	1 |- ((dom A i^i ran B) C_ C -> (A o. B) = U_x e. C ((`'B"{x}) X. (A"{x})))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292   \ cdif 2590   u. cun 2591   i^i cin 2592   C_ wss 2593  (/)c0 2875  {csn 3044  U_ciun 3255   X. cxp 3984  `'ccnv 3985  dom cdm 3986  ran crn 3987  "cima 3989   o. ccom 3990

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-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-rab 2112  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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