Theorem sbthlem7 4439

Description: Lemma for sbth 4443.

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

Proof of Theorem sbthlem7

Step	Hyp	Ref	Expression
1		dmres 3372	. . . . . . . . 9 |- dom ( f |` U.D) = (U.D i^i dom f)
2		inss1 2226	. . . . . . . . 9 |- (U.D i^i dom f) (_ U.D
3	1, 2	eqsstr 2087	. . . . . . . 8 |- dom ( f |` U.D) (_ U.D
4		ssrin 2230	. . . . . . . 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 7	. . . . . . 7 |- (dom ( f |` U.D) i^i dom (`'g |` (A \ U.D))) (_ (U.D i^i dom (`'g |` (A \ U.D)))
6		dmres 3372	. . . . . . . . 9 |- dom (`'g |` (A \ U.D)) = ((A \ U.D) i^i dom `' g)
7		inss1 2226	. . . . . . . . 9 |- ((A \ U.D) i^i dom `' g) (_ (A \ U.D)
8	6, 7	eqsstr 2087	. . . . . . . 8 |- dom (`'g |` (A \ U.D)) (_ (A \ U.D)
9		sslin 2231	. . . . . . . 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 7	. . . . . . 7 |- (U.D i^i dom (`'g |` (A \ U.D))) (_ (U.D i^i (A \ U.D))
11	5, 10	sstri 2069	. . . . . 6 |- (dom ( f |` U.D) i^i dom (`'g |` (A \ U.D))) (_ (U.D i^i (A \ U.D))
12		difdisj 2333	. . . . . 6 |- (U.D i^i (A \ U.D)) = (/)
13	11, 12	sseqtr 2089	. . . . 5 |- (dom ( f |` U.D) i^i dom (`'g |` (A \ U.D))) (_ (/)
14		ss0 2299	. . . . 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 7	. . . 4 |- (dom ( f |` U.D) i^i dom (`'g |` (A \ U.D))) = (/)
16		funun 3546	. . . 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 695	. . 3 |- ((Fun (f |` U.D) /\ Fun (`'g |` (A \ U.D))) -> Fun ((f |` U.D) u. (`'g |` (A \ U.D))))
18		funres 3543	. . 3 |- (Fun f -> Fun (f |` U.D))
19		funres 3543	. . 3 |- (Fun `'g -> Fun (`'g |` (A \ U.D)))
20	17, 18, 19	syl2an 454	. 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 3527	. . 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 7	. 2 |- (Fun H <-> Fun ((f |` U.D) u. (`'g |` (A \ U.D))))
24	20, 23	sylibr 200	1 |- ((Fun f /\ Fun `'g) -> Fun H)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  {cab 1461  Vcvv 1807   \ cdif 2040   u. cun 2041   i^i cin 2042   (_ wss 2043  (/)c0 2276  U.cuni 2498  `'ccnv 3164  dom cdm 3165   |` cres 3167  "cima 3168  Fun wfun 3171

This theorem is referenced by:  sbthlem9 4441

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-res 3185  df-fun 3187

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