Theorem sbthlem8 3356

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
sbthlem8	|- ((Fun `'f /\ (((Fun g /\ dom g = B) /\ ran g (_ A) /\ Fun `'g)) -> Fun `'H)

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

Proof of Theorem sbthlem8

Step	Hyp	Ref	Expression
1		funun 2700	. . 3 |- (((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))))
2		funres11 2709	. . . 4 |- (Fun `'f -> Fun `'(f |` U.D))
3		funcnvcnv 2701	. . . . . . 7 |- (Fun g -> Fun `'`'g)
4		funres11 2709	. . . . . . 7 |- (Fun `'`'g -> Fun `'(`'g |` (A \ U.D)))
5	3, 4	syl 12	. . . . . 6 |- (Fun g -> Fun `'(`'g |` (A \ U.D)))
6	5	adantr 306	. . . . 5 |- ((Fun g /\ dom g = B) -> Fun `'(`'g |` (A \ U.D)))
7	6	ad2antll 320	. . . 4 |- ((((Fun g /\ dom g = B) /\ ran g (_ A) /\ Fun `'g) -> Fun `'(`'g |` (A \ U.D)))
8	2, 7	anim12i 268	. . 3 |- ((Fun `'f /\ (((Fun g /\ dom g = B) /\ ran g (_ A) /\ Fun `'g)) -> (Fun `'(f |` U.D) /\ Fun `'(`'g |` (A \ U.D))))
9		sbthlem.1	. . . . . . . . . 10 |- A e. V
10		sbthlem.2	. . . . . . . . . 10 |- D = {x | (x (_ A /\ (g"(B \ (f"x))) (_ (A \ x))}
11	9, 10	sbthlem4 3352	. . . . . . . . 9 |- (((dom g = B /\ ran g (_ A) /\ Fun `'g) -> (`'g"(A \ U.D)) = (B \ (f"U.D)))
12		df-ima 2431	. . . . . . . . . 10 |- (`'g"(A \ U.D)) = ran (`'g |` (A \ U.D))
13		df-rn 2429	. . . . . . . . . 10 |- ran (`'g |` (A \ U.D)) = dom `'(`'g |` (A \ U.D))
14	12, 13	eqtr 1119	. . . . . . . . 9 |- (`'g"(A \ U.D)) = dom `'(`'g |` (A \ U.D))
15	11, 14	syl5eqr 1138	. . . . . . . 8 |- (((dom g = B /\ ran g (_ A) /\ Fun `'g) -> dom `'(`'g |` (A \ U.D)) = (B \ (f"U.D)))
16		df-ima 2431	. . . . . . . . 9 |- (f"U.D) = ran (f |` U.D)
17		df-rn 2429	. . . . . . . . 9 |- ran (f |` U.D) = dom `'(f |` U.D)
18	16, 17	eqtr2 1120	. . . . . . . 8 |- dom `'(f |` U.D) = (f"U.D)
19	15, 18	jctil 240	. . . . . . 7 |- (((dom g = B /\ ran g (_ A) /\ Fun `'g) -> (dom `'(f |` U.D) = (f"U.D) /\ dom `'(`'g |` (A \ U.D)) = (B \ (f"U.D))))
20		ineq12 1640	. . . . . . 7 |- ((dom `'(f |` U.D) = (f"U.D) /\ dom `'(`'g |` (A \ U.D)) = (B \ (f"U.D))) -> (dom `'(f |` U.D) i^i dom `'(`'g |` (A \ U.D))) = ((f"U.D) i^i (B \ (f"U.D))))
21	19, 20	syl 12	. . . . . 6 |- (((dom g = B /\ ran g (_ A) /\ Fun `'g) -> (dom `'(f |` U.D) i^i dom `'(`'g |` (A \ U.D))) = ((f"U.D) i^i (B \ (f"U.D))))
22		difdisj 1758	. . . . . 6 |- ((f"U.D) i^i (B \ (f"U.D))) = (/)
23	21, 22	syl6eq 1140	. . . . 5 |- (((dom g = B /\ ran g (_ A) /\ Fun `'g) -> (dom `'(f |` U.D) i^i dom `'(`'g |` (A \ U.D))) = (/))
24	23	adantlll 313	. . . 4 |- ((((Fun g /\ dom g = B) /\ ran g (_ A) /\ Fun `'g) -> (dom `'(f |` U.D) i^i dom `'(`'g |` (A \ U.D))) = (/))
25	24	adantl 305	. . 3 |- ((Fun `'f /\ (((Fun g /\ dom g = B) /\ ran g (_ A) /\ Fun `'g)) -> (dom `'(f |` U.D) i^i dom `'(`'g |` (A \ U.D))) = (/))
26	1, 8, 25	sylanc 361	. 2 |- ((Fun `'f /\ (((Fun g /\ dom g = B) /\ ran g (_ A) /\ Fun `'g)) -> Fun (`'(f |` U.D) u. `'(`'g |` (A \ U.D))))
27		sbthlem.3	. . . . 5 |- H = ((f |` U.D) u. (`'g |` (A \ U.D)))
28		cnveq 2513	. . . . 5 |- (H = ((f |` U.D) u. (`'g |` (A \ U.D))) -> `'H = `'((f |` U.D) u. (`'g |` (A \ U.D))))
29	27, 28	ax-mp 6	. . . 4 |- `'H = `'((f |` U.D) u. (`'g |` (A \ U.D)))
30		cnvun 2642	. . . 4 |- `'((f |` U.D) u. (`'g |` (A \ U.D))) = (`'(f |` U.D) u. `'(`'g |` (A \ U.D)))
31	29, 30	eqtr 1119	. . 3 |- `'H = (`'(f |` U.D) u. `'(`'g |` (A \ U.D)))
32		funeq 2683	. . 3 |- (`'H = (`'(f |` U.D) u. `'(`'g |` (A \ U.D))) -> (Fun `'H <-> Fun (`'(f |` U.D) u. `'(`'g |` (A \ U.D)))))
33	31, 32	ax-mp 6	. 2 |- (Fun `'H <-> Fun (`'(f |` U.D) u. `'(`'g |` (A \ U.D))))
34	26, 33	sylibr 175	1 |- ((Fun `'f /\ (((Fun g /\ dom g = B) /\ ran g (_ A) /\ 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  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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