Theorem axdclem 9341

Description: Lemma for axdc 9343. (Contributed by Mario Carneiro, 25-Jan-2013.)

Hypothesis

Ref	Expression
axdclem.1	|- F = ( rec ( ( y e. _V |-> ( g ` { z | y x z } ) ) , s ) |` om )

Assertion

Ref	Expression
axdclem	|- ( ( A. y e. ~P dom x ( y =/= (/) -> ( g ` y ) e. y ) /\ ran x C_ dom x /\ E. z ( F ` K ) x z ) -> ( K e. om -> ( F ` K ) x ( F ` suc K ) ) )

Distinct variable groups:   y, F, z   y, K, z   y, g   y, s   x, y, z

Allowed substitution hints:   F( x, g, s)   K( x, g, s)

Proof of Theorem axdclem

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		neeq1 2856	. . . . . . 7 |- ( y = { z | ( F ` K ) x z } -> ( y =/= (/) <-> { z | ( F ` K ) x z } =/= (/) ) )
2		abn0 3954	. . . . . . 7 |- ( { z | ( F ` K ) x z } =/= (/) <-> E. z ( F ` K ) x z )
3	1, 2	syl6bb 276	. . . . . 6 |- ( y = { z | ( F ` K ) x z } -> ( y =/= (/) <-> E. z ( F ` K ) x z ) )
4		eleq2 2690	. . . . . . . . 9 |- ( y = { z | ( F ` K ) x z } -> ( ( g ` y ) e. y <-> ( g ` y ) e. { z | ( F ` K ) x z } ) )
5		breq2 4657	. . . . . . . . . . 11 |- ( w = z -> ( ( F ` K ) x w <-> ( F ` K ) x z ) )
6	5	cbvabv 2747	. . . . . . . . . 10 |- { w | ( F ` K ) x w } = { z | ( F ` K ) x z }
7	6	eleq2i 2693	. . . . . . . . 9 |- ( ( g ` y ) e. { w | ( F ` K ) x w } <-> ( g ` y ) e. { z | ( F ` K ) x z } )
8	4, 7	syl6bbr 278	. . . . . . . 8 |- ( y = { z | ( F ` K ) x z } -> ( ( g ` y ) e. y <-> ( g ` y ) e. { w | ( F ` K ) x w } ) )
9		fvex 6201	. . . . . . . . 9 |- ( g ` y ) e. _V
10		breq2 4657	. . . . . . . . 9 |- ( w = ( g ` y ) -> ( ( F ` K ) x w <-> ( F ` K ) x ( g ` y ) ) )
11	9, 10	elab 3350	. . . . . . . 8 |- ( ( g ` y ) e. { w | ( F ` K ) x w } <-> ( F ` K ) x ( g ` y ) )
12	8, 11	syl6bb 276	. . . . . . 7 |- ( y = { z | ( F ` K ) x z } -> ( ( g ` y ) e. y <-> ( F ` K ) x ( g ` y ) ) )
13		fveq2 6191	. . . . . . . 8 |- ( y = { z | ( F ` K ) x z } -> ( g ` y ) = ( g ` { z | ( F ` K ) x z } ) )
14	13	breq2d 4665	. . . . . . 7 |- ( y = { z | ( F ` K ) x z } -> ( ( F ` K ) x ( g ` y ) <-> ( F ` K ) x ( g ` { z | ( F ` K ) x z } ) ) )
15	12, 14	bitrd 268	. . . . . 6 |- ( y = { z | ( F ` K ) x z } -> ( ( g ` y ) e. y <-> ( F ` K ) x ( g ` { z | ( F ` K ) x z } ) ) )
16	3, 15	imbi12d 334	. . . . 5 |- ( y = { z | ( F ` K ) x z } -> ( ( y =/= (/) -> ( g ` y ) e. y ) <-> ( E. z ( F ` K ) x z -> ( F ` K ) x ( g ` { z | ( F ` K ) x z } ) ) ) )
17	16	rspcv 3305	. . . 4 |- ( { z | ( F ` K ) x z } e. ~P dom x -> ( A. y e. ~P dom x ( y =/= (/) -> ( g ` y ) e. y ) -> ( E. z ( F ` K ) x z -> ( F ` K ) x ( g ` { z | ( F ` K ) x z } ) ) ) )
18		fvex 6201	. . . . . . . 8 |- ( F ` K ) e. _V
19		vex 3203	. . . . . . . 8 |- z e. _V
20	18, 19	brelrn 5356	. . . . . . 7 |- ( ( F ` K ) x z -> z e. ran x )
21	20	abssi 3677	. . . . . 6 |- { z | ( F ` K ) x z } C_ ran x
22		sstr 3611	. . . . . 6 |- ( ( { z | ( F ` K ) x z } C_ ran x /\ ran x C_ dom x ) -> { z | ( F ` K ) x z } C_ dom x )
23	21, 22	mpan 706	. . . . 5 |- ( ran x C_ dom x -> { z | ( F ` K ) x z } C_ dom x )
24		vex 3203	. . . . . . 7 |- x e. _V
25	24	dmex 7099	. . . . . 6 |- dom x e. _V
26	25	elpw2 4828	. . . . 5 |- ( { z | ( F ` K ) x z } e. ~P dom x <-> { z | ( F ` K ) x z } C_ dom x )
27	23, 26	sylibr 224	. . . 4 |- ( ran x C_ dom x -> { z | ( F ` K ) x z } e. ~P dom x )
28	17, 27	syl11 33	. . 3 |- ( A. y e. ~P dom x ( y =/= (/) -> ( g ` y ) e. y ) -> ( ran x C_ dom x -> ( E. z ( F ` K ) x z -> ( F ` K ) x ( g ` { z | ( F ` K ) x z } ) ) ) )
29	28	3imp 1256	. 2 |- ( ( A. y e. ~P dom x ( y =/= (/) -> ( g ` y ) e. y ) /\ ran x C_ dom x /\ E. z ( F ` K ) x z ) -> ( F ` K ) x ( g ` { z | ( F ` K ) x z } ) )
30		fvex 6201	. . . 4 |- ( g ` { z | ( F ` K ) x z } ) e. _V
31		nfcv 2764	. . . . 5 |- F/_ y s
32		nfcv 2764	. . . . 5 |- F/_ y K
33		nfcv 2764	. . . . 5 |- F/_ y ( g ` { z | ( F ` K ) x z } )
34		axdclem.1	. . . . 5 |- F = ( rec ( ( y e. _V |-> ( g ` { z | y x z } ) ) , s ) |` om )
35		breq1 4656	. . . . . . 7 |- ( y = ( F ` K ) -> ( y x z <-> ( F ` K ) x z ) )
36	35	abbidv 2741	. . . . . 6 |- ( y = ( F ` K ) -> { z | y x z } = { z | ( F ` K ) x z } )
37	36	fveq2d 6195	. . . . 5 |- ( y = ( F ` K ) -> ( g ` { z | y x z } ) = ( g ` { z | ( F ` K ) x z } ) )
38	31, 32, 33, 34, 37	frsucmpt 7533	. . . 4 |- ( ( K e. om /\ ( g ` { z | ( F ` K ) x z } ) e. _V ) -> ( F ` suc K ) = ( g ` { z | ( F ` K ) x z } ) )
39	30, 38	mpan2 707	. . 3 |- ( K e. om -> ( F ` suc K ) = ( g ` { z | ( F ` K ) x z } ) )
40	39	breq2d 4665	. 2 |- ( K e. om -> ( ( F ` K ) x ( F ` suc K ) <-> ( F ` K ) x ( g ` { z | ( F ` K ) x z } ) ) )
41	29, 40	syl5ibrcom 237	1 |- ( ( A. y e. ~P dom x ( y =/= (/) -> ( g ` y ) e. y ) /\ ran x C_ dom x /\ E. z ( F ` K ) x z ) -> ( K e. om -> ( F ` K ) x ( F ` suc K ) ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   |` cres 5116   suc csuc 5725   ` cfv 5888   omcom 7065   reccrdg 7505

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-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-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506

This theorem is referenced by:  axdclem2  9342