Theorem sbthlem4 3352

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))}

Assertion

Ref	Expression
sbthlem4	|- (((dom g = B /\ ran g (_ A) /\ Fun `'g) -> (`'g"(A \ U.D)) = (B \ (f"U.D)))

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

Proof of Theorem sbthlem4

Step	Hyp	Ref	Expression
1		difss 1596	. . . . . . 7 |- (B \ (f"U.D)) (_ B
2		sseq2 1522	. . . . . . 7 |- (dom g = B -> ((B \ (f"U.D)) (_ dom g <-> (B \ (f"U.D)) (_ B))
3	1, 2	mpbiri 169	. . . . . 6 |- (dom g = B -> (B \ (f"U.D)) (_ dom g)
4		ssdmres 2585	. . . . . 6 |- ((B \ (f"U.D)) (_ dom g <-> dom (g |` (B \ (f"U.D))) = (B \ (f"U.D)))
5	3, 4	sylib 173	. . . . 5 |- (dom g = B -> dom (g |` (B \ (f"U.D))) = (B \ (f"U.D)))
6		dfdm4 2525	. . . . 5 |- dom (g |` (B \ (f"U.D))) = ran `'(g |` (B \ (f"U.D)))
7	5, 6	syl5reqr 1139	. . . 4 |- (dom g = B -> (B \ (f"U.D)) = ran `'(g |` (B \ (f"U.D))))
8		funcnvres 2710	. . . . . 6 |- (Fun `'g -> `'(g |` (B \ (f"U.D))) = (`'g |` (g"(B \ (f"U.D)))))
9		sbthlem.1	. . . . . . . 8 |- A e. V
10		sbthlem.2	. . . . . . . 8 |- D = {x | (x (_ A /\ (g"(B \ (f"x))) (_ (A \ x))}
11	9, 10	sbthlem3 3351	. . . . . . 7 |- (ran g (_ A -> (g"(B \ (f"U.D))) = (A \ U.D))
12		reseq2 2576	. . . . . . 7 |- ((g"(B \ (f"U.D))) = (A \ U.D) -> (`'g |` (g"(B \ (f"U.D)))) = (`'g |` (A \ U.D)))
13	11, 12	syl 12	. . . . . 6 |- (ran g (_ A -> (`'g |` (g"(B \ (f"U.D)))) = (`'g |` (A \ U.D)))
14	8, 13	sylan9eqr 1145	. . . . 5 |- ((ran g (_ A /\ Fun `'g) -> `'(g |` (B \ (f"U.D))) = (`'g |` (A \ U.D)))
15	14	rneqd 2557	. . . 4 |- ((ran g (_ A /\ Fun `'g) -> ran `'(g |` (B \ (f"U.D))) = ran (`'g |` (A \ U.D)))
16	7, 15	sylan9eq 1144	. . 3 |- ((dom g = B /\ (ran g (_ A /\ Fun `'g)) -> (B \ (f"U.D)) = ran (`'g |` (A \ U.D)))
17	16	anassrs 338	. 2 |- (((dom g = B /\ ran g (_ A) /\ Fun `'g) -> (B \ (f"U.D)) = ran (`'g |` (A \ U.D)))
18		df-ima 2431	. 2 |- (`'g"(A \ U.D)) = ran (`'g |` (A \ U.D))
19	17, 18	syl6reqr 1143	1 |- (((dom g = B /\ ran g (_ A) /\ Fun `'g) -> (`'g"(A \ U.D)) = (B \ (f"U.D)))

Syntax hints:   -> wi 2   /\ wa 196  {cab 1090   = wceq 1091   e. wcel 1092  Vcvv 1348   \ cdif 1484   (_ 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:  sbthlem6 3354  sbthlem8 3356

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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