Theorem sbthlem8 4434

Description: Lemma for sbth 4437.

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

Proof of Theorem sbthlem8

Step	Hyp	Ref	Expression
1		funun 3540	. . 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 3553	. . . 4 |- (Fun `'f -> Fun `'(f |` U.D))
3		funcnvcnv 3541	. . . . . . 7 |- (Fun g -> Fun `'`'g)
4		funres11 3553	. . . . . . 7 |- (Fun `'`'g -> Fun `'(`'g |` (A \ U.D)))
5	3, 4	syl 10	. . . . . 6 |- (Fun g -> Fun `'(`'g |` (A \ U.D)))
6	5	adantr 389	. . . . 5 |- ((Fun g /\ dom g = B) -> Fun `'(`'g |` (A \ U.D)))
7	6	ad2antrr 404	. . . 4 |- ((((Fun g /\ dom g = B) /\ ran g (_ A) /\ Fun `'g) -> Fun `'(`'g |` (A \ U.D)))
8	2, 7	anim12i 333	. . 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 4430	. . . . . . . . 9 |- (((dom g = B /\ ran g (_ A) /\ Fun `'g) -> (`'g"(A \ U.D)) = (B \ (f"U.D)))
12		df-ima 3181	. . . . . . . . . 10 |- (`'g"(A \ U.D)) = ran (`'g |` (A \ U.D))
13		df-rn 3179	. . . . . . . . . 10 |- ran (`'g |` (A \ U.D)) = dom `'(`'g |` (A \ U.D))
14	12, 13	eqtr 1487	. . . . . . . . 9 |- (`'g"(A \ U.D)) = dom `'(`'g |` (A \ U.D))
15	11, 14	syl5eqr 1513	. . . . . . . 8 |- (((dom g = B /\ ran g (_ A) /\ Fun `'g) -> dom `'(`'g |` (A \ U.D)) = (B \ (f"U.D)))
16		df-ima 3181	. . . . . . . . 9 |- (f"U.D) = ran ( f |` U.D)
17		df-rn 3179	. . . . . . . . 9 |- ran ( f |` U.D) = dom `'(f |` U.D)
18	16, 17	eqtr2 1488	. . . . . . . 8 |- dom `'(f |` U.D) = (f"U.D)
19	15, 18	jctil 292	. . . . . . 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 2202	. . . . . . 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 10	. . . . . 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 2327	. . . . . 6 |- ((f"U.D) i^i (B \ (f"U.D))) = (/)
23	21, 22	syl6eq 1515	. . . . 5 |- (((dom g = B /\ ran g (_ A) /\ Fun `'g) -> (dom `'(f |` U.D) i^i dom `'(`'g |` (A \ U.D))) = (/))
24	23	adantlll 396	. . . 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 388	. . 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 471	. 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 3281	. . . . 5 |- (H = ((f |` U.D) u. (`'g |` (A \ U.D))) -> `'H = `'((f |` U.D) u. (`'g |` (A \ U.D))))
29	27, 28	ax-mp 7	. . . 4 |- `'H = `'((f |` U.D) u. (`'g |` (A \ U.D)))
30		cnvun 3441	. . . 4 |- `'((f |` U.D) u. (`'g |` (A \ U.D))) = (`'(f |` U.D) u. `'(`'g |` (A \ U.D)))
31	29, 30	eqtr 1487	. . 3 |- `'H = (`'(f |` U.D) u. `'(`'g |` (A \ U.D)))
32		funeq 3521	. . 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 7	. 2 |- (Fun `'H <-> Fun (`'(f |` U.D) u. `'(`'g |` (A \ U.D))))
34	26, 33	sylibr 200	1 |- ((Fun `'f /\ (((Fun g /\ dom g = B) /\ ran g (_ A) /\ Fun `'g)) -> Fun `'H)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  {cab 1456  Vcvv 1802   \ cdif 2034   u. cun 2035   i^i cin 2036   (_ wss 2037  (/)c0 2270  U.cuni 2493  `'ccnv 3159  dom cdm 3160  ran crn 3161   |` cres 3162  "cima 3163  Fun wfun 3166

This theorem is referenced by:  sbthlem9 4435

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182

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