Theorem sbthlemi8 7274

Description: Lemma for isbth 7277. (Contributed by NM, 27-Mar-1998.)

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
sbthlemi8	|- ( ( (EXMID /\ 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

Allowed substitution hints:   A( f, g)   B( f, g)   D( f, g)   H( f, g)

Proof of Theorem sbthlemi8

Step	Hyp	Ref	Expression
1		funres11 5451	. . . 4 |- ( Fun `' f -> Fun `' ( f |` U. D ) )
2	1	ad2antlr 493	. . 3 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' ( f |` U. D ) )
3		funcnvcnv 5438	. . . . . 6 |- ( Fun g -> Fun `' `' g )
4		funres11 5451	. . . . . 6 |- ( Fun `' `' g -> Fun `' ( `' g |` ( A \ U. D ) ) )
5	3, 4	syl 14	. . . . 5 |- ( Fun g -> Fun `' ( `' g |` ( A \ U. D ) ) )
6	5	ad2antrr 492	. . . 4 |- ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) -> Fun `' ( `' g |` ( A \ U. D ) ) )
7	6	ad2antrl 494	. . 3 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' ( `' g |` ( A \ U. D ) ) )
8		simpll 531	. . . 4 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> EXMID )
9		simprll 543	. . . . 5 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> ( Fun g /\ dom g = B ) )
10	9	simprd 114	. . . 4 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> dom g = B )
11		simprlr 544	. . . 4 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> ran g C_ A )
12		simprr 537	. . . 4 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' g )
13		df-ima 4785	. . . . . . 7 |- ( f " U. D ) = ran ( f |` U. D )
14		df-rn 4783	. . . . . . 7 |- ran ( f |` U. D ) = dom `' ( f |` U. D )
15	13, 14	eqtr2i 2260	. . . . . 6 |- dom `' ( f |` U. D ) = ( f " U. D )
16		df-ima 4785	. . . . . . . 8 |- ( `' g " ( A \ U. D ) ) = ran ( `' g |` ( A \ U. D ) )
17		df-rn 4783	. . . . . . . 8 |- ran ( `' g |` ( A \ U. D ) ) = dom `' ( `' g |` ( A \ U. D ) )
18	16, 17	eqtri 2259	. . . . . . 7 |- ( `' g " ( A \ U. D ) ) = dom `' ( `' g |` ( A \ U. D ) )
19		sbthlem.1	. . . . . . . 8 |- A e. _V
20		sbthlem.2	. . . . . . . 8 |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }
21	19, 20	sbthlemi4 7270	. . . . . . 7 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( `' g " ( A \ U. D ) ) = ( B \ ( f " U. D ) ) )
22	18, 21	eqtr3id 2285	. . . . . 6 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> dom `' ( `' g |` ( A \ U. D ) ) = ( B \ ( f " U. D ) ) )
23		ineq12 3427	. . . . . 6 |- ( ( 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 ) ) ) )
24	15, 22, 23	sylancr 418	. . . . 5 |- ( (EXMID /\ ( 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 ) ) ) )
25		disjdif 3599	. . . . 5 |- ( ( f " U. D ) i^i ( B \ ( f " U. D ) ) ) = (/)
26	24, 25	eqtrdi 2287	. . . 4 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( dom `' ( f |` U. D ) i^i dom `' ( `' g |` ( A \ U. D ) ) ) = (/) )
27	8, 10, 11, 12, 26	syl121anc 1283	. . 3 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> ( dom `' ( f |` U. D ) i^i dom `' ( `' g |` ( A \ U. D ) ) ) = (/) )
28		funun 5420	. . 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 ) ) ) )
29	2, 7, 27, 28	syl21anc 1277	. 2 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun ( `' ( f |` U. D ) u. `' ( `' g |` ( A \ U. D ) ) ) )
30		sbthlem.3	. . . . 5 |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )
31	30	cnveqi 4953	. . . 4 |- `' H = `' ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )
32		cnvun 5191	. . . 4 |- `' ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) ) = ( `' ( f |` U. D ) u. `' ( `' g |` ( A \ U. D ) ) )
33	31, 32	eqtri 2259	. . 3 |- `' H = ( `' ( f |` U. D ) u. `' ( `' g |` ( A \ U. D ) ) )
34	33	funeqi 5396	. 2 |- ( Fun `' H <-> Fun ( `' ( f |` U. D ) u. `' ( `' g |` ( A \ U. D ) ) ) )
35	29, 34	sylibr 134	1 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' H )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   {cab 2224   _Vcvv 2821   \ cdif 3217   u. cun 3218   i^i cin 3219   C_ wss 3220   (/)c0 3520   U.cuni 3933  EXMIDwem 4329   `'ccnv 4771   dom cdm 4772   ran crn 4773   |` cres 4774   "cima 4775   Fun wfun 5369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-exmid 4330  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-fun 5377

This theorem is referenced by:  sbthlemi9  7275