Theorem sbthlem6 3354

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
sbthlem6	|- ((ran f (_ B /\ ((dom g = B /\ ran g (_ A) /\ Fun `'g)) -> ran H = B)

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

Proof of Theorem sbthlem6

Step	Hyp	Ref	Expression
1		sbthlem.1	. . . . . 6 |- A e. V
2		sbthlem.2	. . . . . 6 |- D = {x | (x (_ A /\ (g"(B \ (f"x))) (_ (A \ x))}
3	1, 2	sbthlem4 3352	. . . . 5 |- (((dom g = B /\ ran g (_ A) /\ Fun `'g) -> (`'g"(A \ U.D)) = (B \ (f"U.D)))
4		df-ima 2431	. . . . 5 |- (`'g"(A \ U.D)) = ran (`'g |` (A \ U.D))
5	3, 4	syl5reqr 1139	. . . 4 |- (((dom g = B /\ ran g (_ A) /\ Fun `'g) -> (B \ (f"U.D)) = ran (`'g |` (A \ U.D)))
6	5	uneq2d 1611	. . 3 |- (((dom g = B /\ ran g (_ A) /\ Fun `'g) -> ((f"U.D) u. (B \ (f"U.D))) = ((f"U.D) u. ran (`'g |` (A \ U.D))))
7		rnun 2644	. . . 4 |- ran ((f |` U.D) u. (`'g |` (A \ U.D))) = (ran (f |` U.D) u. ran (`'g |` (A \ U.D)))
8		sbthlem.3	. . . . 5 |- H = ((f |` U.D) u. (`'g |` (A \ U.D)))
9	8	rneqi 2556	. . . 4 |- ran H = ran ((f |` U.D) u. (`'g |` (A \ U.D)))
10		df-ima 2431	. . . . 5 |- (f"U.D) = ran (f |` U.D)
11	10	uneq1i 1607	. . . 4 |- ((f"U.D) u. ran (`'g |` (A \ U.D))) = (ran (f |` U.D) u. ran (`'g |` (A \ U.D)))
12	7, 9, 11	3eqtr4 1126	. . 3 |- ran H = ((f"U.D) u. ran (`'g |` (A \ U.D)))
13	6, 12	syl6reqr 1143	. 2 |- (((dom g = B /\ ran g (_ A) /\ Fun `'g) -> ran H = ((f"U.D) u. (B \ (f"U.D))))
14		imassrn 2611	. . . 4 |- (f"U.D) (_ ran f
15		sstr2 1510	. . . 4 |- ((f"U.D) (_ ran f -> (ran f (_ B -> (f"U.D) (_ B))
16	14, 15	ax-mp 6	. . 3 |- (ran f (_ B -> (f"U.D) (_ B)
17		ssundif 1764	. . 3 |- ((f"U.D) (_ B <-> ((f"U.D) u. (B \ (f"U.D))) = B)
18	16, 17	sylib 173	. 2 |- (ran f (_ B -> ((f"U.D) u. (B \ (f"U.D))) = B)
19	13, 18	sylan9eqr 1145	1 |- ((ran f (_ B /\ ((dom g = B /\ ran g (_ A) /\ Fun `'g)) -> ran H = B)

Syntax hints:   -> wi 2   /\ wa 196  {cab 1090   = wceq 1091   e. wcel 1092  Vcvv 1348   \ cdif 1484   u. cun 1485   (_ wss 1487  U.cuni 1919  `'ccnv 2409  dom cdm 2410  ran crn 2411   |` cres 2412  "cima 2413  Fun wfun 2416

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-eu 1009  df-mo 1010  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-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432

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