Theorem sbthlem7 3355

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
sbthlem7	|- ((Fun f /\ Fun `'g) -> Fun H)

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

Proof of Theorem sbthlem7

Step	Hyp	Ref	Expression
1		dmres 2584	. . . . . . . . 9 |- dom (f |` U.D) = (U.D i^i dom f)
2		inss1 1657	. . . . . . . . 9 |- (U.D i^i dom f) (_ U.D
3	1, 2	eqsstr 1530	. . . . . . . 8 |- dom (f |` U.D) (_ U.D
4		ssrin 1661	. . . . . . . 8 |- (dom (f |` U.D) (_ U.D -> (dom (f |` U.D) i^i dom (`'g |` (A \ U.D))) (_ (U.D i^i dom (`'g |` (A \ U.D))))
5	3, 4	ax-mp 6	. . . . . . 7 |- (dom (f |` U.D) i^i dom (`'g |` (A \ U.D))) (_ (U.D i^i dom (`'g |` (A \ U.D)))
6		dmres 2584	. . . . . . . . 9 |- dom (`'g |` (A \ U.D)) = ((A \ U.D) i^i dom `'g)
7		inss1 1657	. . . . . . . . 9 |- ((A \ U.D) i^i dom `'g) (_ (A \ U.D)
8	6, 7	eqsstr 1530	. . . . . . . 8 |- dom (`'g |` (A \ U.D)) (_ (A \ U.D)
9		sslin 1662	. . . . . . . 8 |- (dom (`'g |` (A \ U.D)) (_ (A \ U.D) -> (U.D i^i dom (`'g |` (A \ U.D))) (_ (U.D i^i (A \ U.D)))
10	8, 9	ax-mp 6	. . . . . . 7 |- (U.D i^i dom (`'g |` (A \ U.D))) (_ (U.D i^i (A \ U.D))
11	5, 10	sstri 1512	. . . . . 6 |- (dom (f |` U.D) i^i dom (`'g |` (A \ U.D))) (_ (U.D i^i (A \ U.D))
12		difdisj 1758	. . . . . 6 |- (U.D i^i (A \ U.D)) = (/)
13	11, 12	sseqtr 1532	. . . . 5 |- (dom (f |` U.D) i^i dom (`'g |` (A \ U.D))) (_ (/)
14		ss0 1727	. . . . 5 |- ((dom (f |` U.D) i^i dom (`'g |` (A \ U.D))) (_ (/) -> (dom (f |` U.D) i^i dom (`'g |` (A \ U.D))) = (/))
15	13, 14	ax-mp 6	. . . 4 |- (dom (f |` U.D) i^i dom (`'g |` (A \ U.D))) = (/)
16		funun 2700	. . . 4 |- (((Fun (f |` U.D) /\ Fun (`'g |` (A \ U.D))) /\ (dom (f |` U.D) i^i dom (`'g |` (A \ U.D))) = (/)) -> Fun ((f |` U.D) u. (`'g |` (A \ U.D))))
17	15, 16	mpan2 519	. . 3 |- ((Fun (f |` U.D) /\ Fun (`'g |` (A \ U.D))) -> Fun ((f |` U.D) u. (`'g |` (A \ U.D))))
18		funres 2697	. . 3 |- (Fun f -> Fun (f |` U.D))
19		funres 2697	. . 3 |- (Fun `'g -> Fun (`'g |` (A \ U.D)))
20	17, 18, 19	syl2an 349	. 2 |- ((Fun f /\ Fun `'g) -> Fun ((f |` U.D) u. (`'g |` (A \ U.D))))
21		sbthlem.3	. . 3 |- H = ((f |` U.D) u. (`'g |` (A \ U.D)))
22		funeq 2683	. . 3 |- (H = ((f |` U.D) u. (`'g |` (A \ U.D))) -> (Fun H <-> Fun ((f |` U.D) u. (`'g |` (A \ U.D)))))
23	21, 22	ax-mp 6	. 2 |- (Fun H <-> Fun ((f |` U.D) u. (`'g |` (A \ U.D))))
24	20, 23	sylibr 175	1 |- ((Fun f /\ Fun `'g) -> Fun H)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  {cab 1090   = wceq 1091   e. wcel 1092  Vcvv 1348   \ cdif 1484   u. cun 1485   i^i cin 1486   (_ wss 1487  (/)c0 1707  U.cuni 1919  `'ccnv 2409  dom cdm 2410   |` 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-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-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-res 2430  df-fun 2432

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