Theorem sbthlem8 5517

Description: Lemma for sbth 5520.

Hypotheses

Ref	Expression
sbthlem.1	|- A e. _V
sbthlem.2	|- D = {x | (x C_ A /\ (g"(B \ (f"x))) C_ (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 C_ A) /\ Fun `'g)) -> Fun `'H)

Distinct variable groups:   x,A   x,B   x,D   x,f   x,g   x,H

Proof of Theorem sbthlem8

Step	Hyp	Ref	Expression
1		funres11 4486	. . . 4 |- (Fun `'f -> Fun `'(f |` U.D))
2		funcnvcnv 4473	. . . . . . 7 |- (Fun g -> Fun `'`'g)
3		funres11 4486	. . . . . . 7 |- (Fun `'`'g -> Fun `'(`'g |` (A \ U.D)))
4	2, 3	syl 12	. . . . . 6 |- (Fun g -> Fun `'(`'g |` (A \ U.D)))
5	4	adantr 425	. . . . 5 |- ((Fun g /\ dom g = B) -> Fun `'(`'g |` (A \ U.D)))
6	5	ad2antrr 440	. . . 4 |- ((((Fun g /\ dom g = B) /\ ran g C_ A) /\ Fun `'g) -> Fun `'(`'g |` (A \ U.D)))
7	1, 6	anim12i 360	. . 3 |- ((Fun `'f /\ (((Fun g /\ dom g = B) /\ ran g C_ A) /\ Fun `'g)) -> (Fun `'(f |` U.D) /\ Fun `'(`'g |` (A \ U.D))))
8		ineq12 2791	. . . . . . 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))))
9		df-ima 4007	. . . . . . . 8 |- (f"U.D) = ran ( f |` U.D)
10		df-rn 4005	. . . . . . . 8 |- ran ( f |` U.D) = dom `'(f |` U.D)
11	9, 10	eqtr2i 1909	. . . . . . 7 |- dom `'(f |` U.D) = (f"U.D)
12		sbthlem.1	. . . . . . . . 9 |- A e. _V
13		sbthlem.2	. . . . . . . . 9 |- D = {x | (x C_ A /\ (g"(B \ (f"x))) C_ (A \ x))}
14	12, 13	sbthlem4 5513	. . . . . . . 8 |- (((dom g = B /\ ran g C_ A) /\ Fun `'g) -> (`'g"(A \ U.D)) = (B \ (f"U.D)))
15		df-ima 4007	. . . . . . . . 9 |- (`'g"(A \ U.D)) = ran (`'g |` (A \ U.D))
16		df-rn 4005	. . . . . . . . 9 |- ran (`'g |` (A \ U.D)) = dom `'(`'g |` (A \ U.D))
17	15, 16	eqtri 1908	. . . . . . . 8 |- (`'g"(A \ U.D)) = dom `'(`'g |` (A \ U.D))
18	14, 17	syl5eqr 1942	. . . . . . 7 |- (((dom g = B /\ ran g C_ A) /\ Fun `'g) -> dom `'(`'g |` (A \ U.D)) = (B \ (f"U.D)))
19	8, 11, 18	sylancr 526	. . . . . 6 |- (((dom g = B /\ ran g C_ A) /\ Fun `'g) -> (dom `'(f |` U.D) i^i dom `'(`'g |` (A \ U.D))) = ((f"U.D) i^i (B \ (f"U.D))))
20		difdisj 2945	. . . . . 6 |- ((f"U.D) i^i (B \ (f"U.D))) = (/)
21	19, 20	syl6eq 1944	. . . . 5 |- (((dom g = B /\ ran g C_ A) /\ Fun `'g) -> (dom `'(f |` U.D) i^i dom `'(`'g |` (A \ U.D))) = (/))
22	21	adantlll 432	. . . 4 |- ((((Fun g /\ dom g = B) /\ ran g C_ A) /\ Fun `'g) -> (dom `'(f |` U.D) i^i dom `'(`'g |` (A \ U.D))) = (/))
23	22	adantl 424	. . 3 |- ((Fun `'f /\ (((Fun g /\ dom g = B) /\ ran g C_ A) /\ Fun `'g)) -> (dom `'(f |` U.D) i^i dom `'(`'g |` (A \ U.D))) = (/))
24		funun 4462	. . 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))))
25	7, 23, 24	syl11anc 524	. 2 |- ((Fun `'f /\ (((Fun g /\ dom g = B) /\ ran g C_ A) /\ Fun `'g)) -> Fun (`'(f |` U.D) u. `'(`'g |` (A \ U.D))))
26		sbthlem.3	. . . . 5 |- H = ((f |` U.D) u. (`'g |` (A \ U.D)))
27	26	cnveqi 4136	. . . 4 |- `'H = `'((f |` U.D) u. (`'g |` (A \ U.D)))
28		cnvun 4322	. . . 4 |- `'((f |` U.D) u. (`'g |` (A \ U.D))) = (`'(f |` U.D) u. `'(`'g |` (A \ U.D)))
29	27, 28	eqtri 1908	. . 3 |- `'H = (`'(f |` U.D) u. `'(`'g |` (A \ U.D)))
30		funeq 4441	. . 3 |- (`'H = (`'(f |` U.D) u. `'(`'g |` (A \ U.D))) -> (Fun `'H <-> Fun (`'(f |` U.D) u. `'(`'g |` (A \ U.D)))))
31	29, 30	ax-mp 7	. 2 |- (Fun `'H <-> Fun (`'(f |` U.D) u. `'(`'g |` (A \ U.D))))
32	25, 31	sylibr 217	1 |- ((Fun `'f /\ (((Fun g /\ dom g = B) /\ ran g C_ A) /\ Fun `'g)) -> Fun `'H)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  _Vcvv 2292   \ cdif 2590   u. cun 2591   i^i cin 2592   C_ wss 2593  (/)c0 2875  U.cuni 3177  `'ccnv 3985  dom cdm 3986  ran crn 3987   |` cres 3988  "cima 3989  Fun wfun 3992

This theorem is referenced by:  sbthlem9 5518

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008

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