Theorem pwfilem 8260

Description: Lemma for pwfi 8261. (Contributed by NM, 26-Mar-2007.)

Hypothesis

Ref	Expression
pwfilem.1	|- F = ( c e. ~P b |-> ( c u. { x } ) )

Assertion

Ref	Expression
pwfilem	|- ( ~P b e. Fin -> ~P ( b u. { x } ) e. Fin )

Distinct variable groups:   b, c   x, c

Allowed substitution hints:   F( x, b, c)

Proof of Theorem pwfilem

Dummy variable d is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwundif 5021	. 2 |- ~P ( b u. { x } ) = ( ( ~P ( b u. { x } ) \ ~P b ) u. ~P b )
2		vex 3203	. . . . . . . . 9 |- c e. _V
3		snex 4908	. . . . . . . . 9 |- { x } e. _V
4	2, 3	unex 6956	. . . . . . . 8 |- ( c u. { x } ) e. _V
5		pwfilem.1	. . . . . . . 8 |- F = ( c e. ~P b |-> ( c u. { x } ) )
6	4, 5	fnmpti 6022	. . . . . . 7 |- F Fn ~P b
7		dffn4 6121	. . . . . . 7 |- ( F Fn ~P b <-> F : ~P b -onto-> ran F )
8	6, 7	mpbi 220	. . . . . 6 |- F : ~P b -onto-> ran F
9		fodomfi 8239	. . . . . 6 |- ( ( ~P b e. Fin /\ F : ~P b -onto-> ran F ) -> ran F ~<_ ~P b )
10	8, 9	mpan2 707	. . . . 5 |- ( ~P b e. Fin -> ran F ~<_ ~P b )
11		domfi 8181	. . . . 5 |- ( ( ~P b e. Fin /\ ran F ~<_ ~P b ) -> ran F e. Fin )
12	10, 11	mpdan 702	. . . 4 |- ( ~P b e. Fin -> ran F e. Fin )
13		eldifi 3732	. . . . . . . . 9 |- ( d e. ( ~P ( b u. { x } ) \ ~P b ) -> d e. ~P ( b u. { x } ) )
14	3	elpwun 6977	. . . . . . . . 9 |- ( d e. ~P ( b u. { x } ) <-> ( d \ { x } ) e. ~P b )
15	13, 14	sylib 208	. . . . . . . 8 |- ( d e. ( ~P ( b u. { x } ) \ ~P b ) -> ( d \ { x } ) e. ~P b )
16		undif1 4043	. . . . . . . . 9 |- ( ( d \ { x } ) u. { x } ) = ( d u. { x } )
17		elpwunsn 4224	. . . . . . . . . . 11 |- ( d e. ( ~P ( b u. { x } ) \ ~P b ) -> x e. d )
18	17	snssd 4340	. . . . . . . . . 10 |- ( d e. ( ~P ( b u. { x } ) \ ~P b ) -> { x } C_ d )
19		ssequn2 3786	. . . . . . . . . 10 |- ( { x } C_ d <-> ( d u. { x } ) = d )
20	18, 19	sylib 208	. . . . . . . . 9 |- ( d e. ( ~P ( b u. { x } ) \ ~P b ) -> ( d u. { x } ) = d )
21	16, 20	syl5req 2669	. . . . . . . 8 |- ( d e. ( ~P ( b u. { x } ) \ ~P b ) -> d = ( ( d \ { x } ) u. { x } ) )
22		uneq1 3760	. . . . . . . . . 10 |- ( c = ( d \ { x } ) -> ( c u. { x } ) = ( ( d \ { x } ) u. { x } ) )
23	22	eqeq2d 2632	. . . . . . . . 9 |- ( c = ( d \ { x } ) -> ( d = ( c u. { x } ) <-> d = ( ( d \ { x } ) u. { x } ) ) )
24	23	rspcev 3309	. . . . . . . 8 |- ( ( ( d \ { x } ) e. ~P b /\ d = ( ( d \ { x } ) u. { x } ) ) -> E. c e. ~P b d = ( c u. { x } ) )
25	15, 21, 24	syl2anc 693	. . . . . . 7 |- ( d e. ( ~P ( b u. { x } ) \ ~P b ) -> E. c e. ~P b d = ( c u. { x } ) )
26	5, 4	elrnmpti 5376	. . . . . . 7 |- ( d e. ran F <-> E. c e. ~P b d = ( c u. { x } ) )
27	25, 26	sylibr 224	. . . . . 6 |- ( d e. ( ~P ( b u. { x } ) \ ~P b ) -> d e. ran F )
28	27	ssriv 3607	. . . . 5 |- ( ~P ( b u. { x } ) \ ~P b ) C_ ran F
29		ssdomg 8001	. . . . 5 |- ( ran F e. Fin -> ( ( ~P ( b u. { x } ) \ ~P b ) C_ ran F -> ( ~P ( b u. { x } ) \ ~P b ) ~<_ ran F ) )
30	12, 28, 29	mpisyl 21	. . . 4 |- ( ~P b e. Fin -> ( ~P ( b u. { x } ) \ ~P b ) ~<_ ran F )
31		domfi 8181	. . . 4 |- ( ( ran F e. Fin /\ ( ~P ( b u. { x } ) \ ~P b ) ~<_ ran F ) -> ( ~P ( b u. { x } ) \ ~P b ) e. Fin )
32	12, 30, 31	syl2anc 693	. . 3 |- ( ~P b e. Fin -> ( ~P ( b u. { x } ) \ ~P b ) e. Fin )
33		unfi 8227	. . 3 |- ( ( ( ~P ( b u. { x } ) \ ~P b ) e. Fin /\ ~P b e. Fin ) -> ( ( ~P ( b u. { x } ) \ ~P b ) u. ~P b ) e. Fin )
34	32, 33	mpancom 703	. 2 |- ( ~P b e. Fin -> ( ( ~P ( b u. { x } ) \ ~P b ) u. ~P b ) e. Fin )
35	1, 34	syl5eqel 2705	1 |- ( ~P b e. Fin -> ~P ( b u. { x } ) e. Fin )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   E.wrex 2913   \ cdif 3571   u. cun 3572   C_ wss 3574   ~Pcpw 4158   {csn 4177   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   Fn wfn 5883   -onto->wfo 5886   ~<_ cdom 7953   Fincfn 7955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-fin 7959

This theorem is referenced by:  pwfi  8261