Theorem sbthlem5 3353

Description: Lemma for Schroeder-Bernstein Theorem.

Hypotheses

Ref	Expression
sbthlem.1	|- A e. V
sbthlem.2	|- D = {x | (x (_ A /\ (g"(B \ (f"x))) (_ (A \ x))}
sbthlem.3	|- H = ((f |` U.D) u. (`'g |` (A \ U.D)))

Assertion

Ref	Expression
sbthlem5	|- ((dom f = A /\ ran g (_ A) -> dom H = A)

Distinct variable group(s):   x,A   x,B   x,D   x,f   x,g   x,H

Proof of Theorem sbthlem5

Step	Hyp	Ref	Expression
1		sbthlem.1	. . . . . . . . 9 |- A e. V
2		sbthlem.2	. . . . . . . . 9 |- D = {x | (x (_ A /\ (g"(B \ (f"x))) (_ (A \ x))}
3	1, 2	sbthlem1 3349	. . . . . . . 8 |- U.D (_ (A \ (g"(B \ (f"U.D))))
4		difss 1596	. . . . . . . 8 |- (A \ (g"(B \ (f"U.D)))) (_ A
5	3, 4	sstri 1512	. . . . . . 7 |- U.D (_ A
6		sseq2 1522	. . . . . . 7 |- (dom f = A -> (U.D (_ dom f <-> U.D (_ A))
7	5, 6	mpbiri 169	. . . . . 6 |- (dom f = A -> U.D (_ dom f)
8		dfss 1493	. . . . . 6 |- (U.D (_ dom f <-> U.D = (U.D i^i dom f))
9	7, 8	sylib 173	. . . . 5 |- (dom f = A -> U.D = (U.D i^i dom f))
10	9	uneq1d 1610	. . . 4 |- (dom f = A -> (U.D u. (A \ U.D)) = ((U.D i^i dom f) u. (A \ U.D)))
11		imassrn 2611	. . . . . . 7 |- (g"(B \ (f"U.D))) (_ ran g
12	1, 2	sbthlem3 3351	. . . . . . . 8 |- (ran g (_ A -> (g"(B \ (f"U.D))) = (A \ U.D))
13	12	sseq1d 1527	. . . . . . 7 |- (ran g (_ A -> ((g"(B \ (f"U.D))) (_ ran g <-> (A \ U.D) (_ ran g))
14	11, 13	mpbii 168	. . . . . 6 |- (ran g (_ A -> (A \ U.D) (_ ran g)
15		dfss 1493	. . . . . 6 |- ((A \ U.D) (_ ran g <-> (A \ U.D) = ((A \ U.D) i^i ran g))
16	14, 15	sylib 173	. . . . 5 |- (ran g (_ A -> (A \ U.D) = ((A \ U.D) i^i ran g))
17	16	uneq2d 1611	. . . 4 |- (ran g (_ A -> ((U.D i^i dom f) u. (A \ U.D)) = ((U.D i^i dom f) u. ((A \ U.D) i^i ran g)))
18	10, 17	sylan9eq 1144	. . 3 |- ((dom f = A /\ ran g (_ A) -> (U.D u. (A \ U.D)) = ((U.D i^i dom f) u. ((A \ U.D) i^i ran g)))
19		sbthlem.3	. . . . 5 |- H = ((f |` U.D) u. (`'g |` (A \ U.D)))
20	19	dmeqi 2532	. . . 4 |- dom H = dom ((f |` U.D) u. (`'g |` (A \ U.D)))
21		dmun 2536	. . . 4 |- dom ((f |` U.D) u. (`'g |` (A \ U.D))) = (dom (f |` U.D) u. dom (`'g |` (A \ U.D)))
22		dmres 2584	. . . . 5 |- dom (f |` U.D) = (U.D i^i dom f)
23		dmres 2584	. . . . . 6 |- dom (`'g |` (A \ U.D)) = ((A \ U.D) i^i dom `'g)
24		df-rn 2429	. . . . . . . 8 |- ran g = dom `'g
25	24	cleqcomi 1105	. . . . . . 7 |- dom `'g = ran g
26	25	ineq2i 1642	. . . . . 6 |- ((A \ U.D) i^i dom `'g) = ((A \ U.D) i^i ran g)
27	23, 26	eqtr 1119	. . . . 5 |- dom (`'g |` (A \ U.D)) = ((A \ U.D) i^i ran g)
28	22, 27	uneq12i 1609	. . . 4 |- (dom (f |` U.D) u. dom (`'g |` (A \ U.D))) = ((U.D i^i dom f) u. ((A \ U.D) i^i ran g))
29	20, 21, 28	3eqtr 1123	. . 3 |- dom H = ((U.D i^i dom f) u. ((A \ U.D) i^i ran g))
30	18, 29	syl6reqr 1143	. 2 |- ((dom f = A /\ ran g (_ A) -> dom H = (U.D u. (A \ U.D)))
31		ssundif 1764	. . 3 |- (U.D (_ A <-> (U.D u. (A \ U.D)) = A)
32	5, 31	mpbi 164	. 2 |- (U.D u. (A \ U.D)) = A
33	30, 32	syl6eq 1140	1 |- ((dom f = A /\ ran g (_ A) -> dom H = A)

Syntax hints:   -> wi 2   /\ wa 196  {cab 1090   = wceq 1091   e. wcel 1092  Vcvv 1348   \ cdif 1484   u. cun 1485   i^i cin 1486   (_ wss 1487  U.cuni 1919  `'ccnv 2409  dom cdm 2410  ran crn 2411   |` cres 2412  "cima 2413

This theorem is referenced by:  sbthlem9 3357

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431

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