Theorem pwfilem 5660

Description: Lemma for pwfi 5661.

Hypothesis

Ref	Expression
pwfilem.1	|- F = {<.c, y>. | (c e. ~Pb /\ y = (c u. {x}))}

Assertion

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

Distinct variable groups:   y,c,b   x,c,y

Proof of Theorem pwfilem

Step	Hyp	Ref	Expression
1		domfi 5631	. . . 4 |- ((ran F e. Fin /\ (~P(b u. {x}) \ ~Pb) ~<_ ran F) -> (~P(b u. {x}) \ ~Pb) e. Fin)
2		visset 2295	. . . . . . . . 9 |- c e. _V
3		snex 3492	. . . . . . . . 9 |- {x} e. _V
4	2, 3	unex 3796	. . . . . . . 8 |- (c u. {x}) e. _V
5		pwfilem.1	. . . . . . . 8 |- F = {<.c, y>. | (c e. ~Pb /\ y = (c u. {x}))}
6	4, 5	fnopab2 4549	. . . . . . 7 |- F Fn ~Pb
7		dffn4 4623	. . . . . . 7 |- (F Fn ~Pb <-> F:~Pb-onto->ran F)
8	6, 7	mpbi 206	. . . . . 6 |- F:~Pb-onto->ran F
9		fodomfi 5656	. . . . . 6 |- ((~Pb e. Fin /\ F:~Pb-onto->ran F) -> ran F ~<_ ~Pb)
10	8, 9	mpan2 760	. . . . 5 |- (~Pb e. Fin -> ran F ~<_ ~Pb)
11		domfi 5631	. . . . 5 |- ((~Pb e. Fin /\ ran F ~<_ ~Pb) -> ran F e. Fin)
12	10, 11	mpdan 768	. . . 4 |- (~Pb e. Fin -> ran F e. Fin)
13		visset 2295	. . . . . . . 8 |- b e. _V
14	13, 3	unex 3796	. . . . . . 7 |- (b u. {x}) e. _V
15	14	pwex 3487	. . . . . 6 |- ~P(b u. {x}) e. _V
16		difexg 3458	. . . . . 6 |- (~P(b u. {x}) e. _V -> (~P(b u. {x}) \ ~Pb) e. _V)
17	15, 16	ax-mp 7	. . . . 5 |- (~P(b u. {x}) \ ~Pb) e. _V
18		eldifi 2730	. . . . . . . . 9 |- (d e. (~P(b u. {x}) \ ~Pb) -> d e. ~P(b u. {x}))
19	3	elpwun 3855	. . . . . . . . 9 |- (d e. ~P(b u. {x}) <-> (d \ {x}) e. ~Pb)
20	18, 19	sylib 215	. . . . . . . 8 |- (d e. (~P(b u. {x}) \ ~Pb) -> (d \ {x}) e. ~Pb)
21		elpwunsn 3856	. . . . . . . . . . 11 |- (d e. (~P(b u. {x}) \ ~Pb) -> x e. d)
22	21	snssd 3130	. . . . . . . . . 10 |- (d e. (~P(b u. {x}) \ ~Pb) -> {x} C_ d)
23		ssequn2 2779	. . . . . . . . . 10 |- ({x} C_ d <-> (d u. {x}) = d)
24	22, 23	sylib 215	. . . . . . . . 9 |- (d e. (~P(b u. {x}) \ ~Pb) -> (d u. {x}) = d)
25		undif1 2949	. . . . . . . . 9 |- ((d \ {x}) u. {x}) = (d u. {x})
26	24, 25	syl5req 1941	. . . . . . . 8 |- (d e. (~P(b u. {x}) \ ~Pb) -> d = ((d \ {x}) u. {x}))
27		uneq1 2748	. . . . . . . . . 10 |- (c = (d \ {x}) -> (c u. {x}) = ((d \ {x}) u. {x}))
28	27	eqeq2d 1895	. . . . . . . . 9 |- (c = (d \ {x}) -> (d = (c u. {x}) <-> d = ((d \ {x}) u. {x})))
29	28	rcla4ev 2381	. . . . . . . 8 |- (((d \ {x}) e. ~Pb /\ d = ((d \ {x}) u. {x})) -> E.c e. ~P bd = (c u. {x}))
30	20, 26, 29	syl11anc 524	. . . . . . 7 |- (d e. (~P(b u. {x}) \ ~Pb) -> E.c e. ~P bd = (c u. {x}))
31	4, 5	elrnopab 4774	. . . . . . 7 |- (d e. ran F <-> E.c e. ~P bd = (c u. {x}))
32	30, 31	sylibr 217	. . . . . 6 |- (d e. (~P(b u. {x}) \ ~Pb) -> d e. ran F)
33	32	ssriv 2621	. . . . 5 |- (~P(b u. {x}) \ ~Pb) C_ ran F
34		ssdomg 5467	. . . . 5 |- ((~P(b u. {x}) \ ~Pb) e. _V -> ((~P(b u. {x}) \ ~Pb) C_ ran F -> (~P(b u. {x}) \ ~Pb) ~<_ ran F))
35	17, 33, 34	mp2 54	. . . 4 |- (~P(b u. {x}) \ ~Pb) ~<_ ran F
36	1, 12, 35	sylancl 525	. . 3 |- (~Pb e. Fin -> (~P(b u. {x}) \ ~Pb) e. Fin)
37		unfi 5644	. . 3 |- (((~P(b u. {x}) \ ~Pb) e. Fin /\ ~Pb e. Fin) -> ((~P(b u. {x}) \ ~Pb) u. ~Pb) e. Fin)
38	36, 37	mpancom 769	. 2 |- (~Pb e. Fin -> ((~P(b u. {x}) \ ~Pb) u. ~Pb) e. Fin)
39		pwundif 3579	. . 3 |- ~P(b u. {x}) = ((~P(b u. {x}) \ ~Pb) u. ~Pb)
40	39	eleq1i 1960	. 2 |- (~P(b u. {x}) e. Fin <-> ((~P(b u. {x}) \ ~Pb) u. ~Pb) e. Fin)
41	38, 40	sylibr 217	1 |- (~Pb e. Fin -> ~P(b u. {x}) e. Fin)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E.wrex 2106  _Vcvv 2292   \ cdif 2590   u. cun 2591   C_ wss 2593  ~Pcpw 3032  {csn 3044   class class class wbr 3338  {copab 3395  ran crn 3987   Fn wfn 3993  -onto->wfo 3996   ~<_ cdom 5424  Fincfn 5426

This theorem is referenced by:  pwfi 5661

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-13 1311  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-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  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-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  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  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-rdg 5140  df-1o 5177  df-oadd 5179  df-er 5318  df-en 5427  df-dom 5428  df-fin 5430

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