Theorem bnj75 13310

Description: Technical lemma of bnj69 13455. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem).

Hypotheses

Ref	Expression
bnj75.1	|- (ph <-> (f` (/)) = pred(x, A, R))
bnj75.2	|- (ps <-> A.i e. om (suc i e. n -> (f` suc i) = U_y e. (f` i) pred(y, A, R)))
bnj75.3	|- D = (om \ {(/)})
bnj75.4	|- (ch <-> ((R FrSe A /\ x e. A) -> E!f(f Fn n /\ ph /\ ps)))
bnj75.5	|- (th <-> A.m e. D (m _E n -> [m / n]ch))

Assertion

Ref	Expression
bnj75	|- ((R FrSe A /\ x e. A) -> A.n e. D E!f(f Fn n /\ ph /\ ps))

Distinct variable groups:   A,f,i,m,n,x,y   D,f,i,m,n   R,f,i,m,n,x,y   ch,m   ph,i,m   ps,m

Proof of Theorem bnj75

Step	Hyp	Ref	Expression
1		ax-17 1317	. 2 |- ((R FrSe A /\ x e. A) -> A.n(R FrSe A /\ x e. A))
2		bnj75.5	. . . . 5 |- (th <-> A.m e. D (m _E n -> [m / n]ch))
3		bnj75.3	. . . . . 6 |- D = (om \ {(/)})
4	3	bnj113 12458	. . . . 5 |- D e. _V
5		zfregfr 5706	. . . . 5 |- _E Fr D
6	2, 4, 5	bnj157 13200	. . . 4 |- (A.n e. D (th -> ch) -> A.n e. D ch)
7		bnj75.1	. . . . . . 7 |- (ph <-> (f` (/)) = pred(x, A, R))
8		bnj75.2	. . . . . . 7 |- (ps <-> A.i e. om (suc i e. n -> (f` suc i) = U_y e. (f` i) pred(y, A, R)))
9		bnj75.4	. . . . . . 7 |- (ch <-> ((R FrSe A /\ x e. A) -> E!f(f Fn n /\ ph /\ ps)))
10	7, 8, 3, 9, 2	bnj153 13247	. . . . . 6 |- (n = 1o -> ((n e. D /\ th) -> ch))
11	7, 8, 3, 9, 2	bnj601 13309	. . . . . 6 |- (n =/= 1o -> ((n e. D /\ th) -> ch))
12	10, 11	pm2.61ine 2089	. . . . 5 |- ((n e. D /\ th) -> ch)
13	12	ex 402	. . . 4 |- (n e. D -> (th -> ch))
14	6, 13	mprg 2162	. . 3 |- A.n e. D ch
15	14, 9	bnj614 12567	. 2 |- A.n e. D ((R FrSe A /\ x e. A) -> E!f(f Fn n /\ ph /\ ps))
16	1, 15	bnj586 12552	1 |- ((R FrSe A /\ x e. A) -> A.n e. D E!f(f Fn n /\ ph /\ ps))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  [wsbc 1534  E!weu 1771  A.wral 2105   \ cdif 2590  (/)c0 2875  {csn 3044  U_ciun 3255   class class class wbr 3338   _E cep 3581  suc csuc 3659  omcom 3949   Fn wfn 3993  ` cfv 3998  1oc1o 5172   predsyn-bnj14 12023   FrSe syn-bnj15 12027

This theorem is referenced by:  bnj616 13311

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  ax-reg 5695  ax-inf2 5731

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-nel 2020  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-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-1o 5177  df-bnj17 12020  df-bnj14 12024  df-bnj13 12026  df-bnj15 12028

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