Theorem bnj535 30960

Description: Technical lemma for bnj852 30991. 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). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj535.1	|- ( ph' <-> ( f ` (/) ) = pred ( x , A , R ) )
bnj535.2	|- ( ps' <-> A. i e. om ( suc i e. m -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj535.3	|- G = ( f u. { <. m , U_ y e. ( f ` p ) pred ( y , A , R ) >. } )
bnj535.4	|- ( ta <-> ( ph' /\ ps' /\ m e. om /\ p e. m ) )

Assertion

Ref	Expression
bnj535	|- ( ( R FrSe A /\ ta /\ n = ( m u. { m } ) /\ f Fn m ) -> G Fn n )

Distinct variable groups:   A, i, p, y   R, i, p, y   f, i, p, y   i, m, p   p, ph'

Allowed substitution hints:   ta( x, y, f, i, m, n, p)   A( x, f, m, n)   R( x, f, m, n)   G( x, y, f, i, m, n, p)   ph'( x, y, f, i, m, n)   ps'( x, y, f, i, m, n, p)

Proof of Theorem bnj535

Step	Hyp	Ref	Expression
1		bnj422 30781	. . 3 |- ( ( R FrSe A /\ ta /\ n = ( m u. { m } ) /\ f Fn m ) <-> ( n = ( m u. { m } ) /\ f Fn m /\ R FrSe A /\ ta ) )
2		bnj251 30768	. . 3 |- ( ( n = ( m u. { m } ) /\ f Fn m /\ R FrSe A /\ ta ) <-> ( n = ( m u. { m } ) /\ ( f Fn m /\ ( R FrSe A /\ ta ) ) ) )
3	1, 2	bitri 264	. 2 |- ( ( R FrSe A /\ ta /\ n = ( m u. { m } ) /\ f Fn m ) <-> ( n = ( m u. { m } ) /\ ( f Fn m /\ ( R FrSe A /\ ta ) ) ) )
4		fvex 6201	. . . . . . . . 9 |- ( f ` p ) e. _V
5		bnj535.1	. . . . . . . . . 10 |- ( ph' <-> ( f ` (/) ) = pred ( x , A , R ) )
6		bnj535.2	. . . . . . . . . 10 |- ( ps' <-> A. i e. om ( suc i e. m -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
7		bnj535.4	. . . . . . . . . 10 |- ( ta <-> ( ph' /\ ps' /\ m e. om /\ p e. m ) )
8	5, 6, 7	bnj518 30956	. . . . . . . . 9 |- ( ( R FrSe A /\ ta ) -> A. y e. ( f ` p ) pred ( y , A , R ) e. _V )
9		iunexg 7143	. . . . . . . . 9 |- ( ( ( f ` p ) e. _V /\ A. y e. ( f ` p ) pred ( y , A , R ) e. _V ) -> U_ y e. ( f ` p ) pred ( y , A , R ) e. _V )
10	4, 8, 9	sylancr 695	. . . . . . . 8 |- ( ( R FrSe A /\ ta ) -> U_ y e. ( f ` p ) pred ( y , A , R ) e. _V )
11		vex 3203	. . . . . . . . 9 |- m e. _V
12	11	bnj519 30804	. . . . . . . 8 |- ( U_ y e. ( f ` p ) pred ( y , A , R ) e. _V -> Fun { <. m , U_ y e. ( f ` p ) pred ( y , A , R ) >. } )
13	10, 12	syl 17	. . . . . . 7 |- ( ( R FrSe A /\ ta ) -> Fun { <. m , U_ y e. ( f ` p ) pred ( y , A , R ) >. } )
14		dmsnopg 5606	. . . . . . . 8 |- ( U_ y e. ( f ` p ) pred ( y , A , R ) e. _V -> dom { <. m , U_ y e. ( f ` p ) pred ( y , A , R ) >. } = { m } )
15	10, 14	syl 17	. . . . . . 7 |- ( ( R FrSe A /\ ta ) -> dom { <. m , U_ y e. ( f ` p ) pred ( y , A , R ) >. } = { m } )
16	13, 15	bnj1422 30908	. . . . . 6 |- ( ( R FrSe A /\ ta ) -> { <. m , U_ y e. ( f ` p ) pred ( y , A , R ) >. } Fn { m } )
17		bnj521 30805	. . . . . . 7 |- ( m i^i { m } ) = (/)
18		fnun 5997	. . . . . . 7 |- ( ( ( f Fn m /\ { <. m , U_ y e. ( f ` p ) pred ( y , A , R ) >. } Fn { m } ) /\ ( m i^i { m } ) = (/) ) -> ( f u. { <. m , U_ y e. ( f ` p ) pred ( y , A , R ) >. } ) Fn ( m u. { m } ) )
19	17, 18	mpan2 707	. . . . . 6 |- ( ( f Fn m /\ { <. m , U_ y e. ( f ` p ) pred ( y , A , R ) >. } Fn { m } ) -> ( f u. { <. m , U_ y e. ( f ` p ) pred ( y , A , R ) >. } ) Fn ( m u. { m } ) )
20	16, 19	sylan2 491	. . . . 5 |- ( ( f Fn m /\ ( R FrSe A /\ ta ) ) -> ( f u. { <. m , U_ y e. ( f ` p ) pred ( y , A , R ) >. } ) Fn ( m u. { m } ) )
21		bnj535.3	. . . . . 6 |- G = ( f u. { <. m , U_ y e. ( f ` p ) pred ( y , A , R ) >. } )
22	21	fneq1i 5985	. . . . 5 |- ( G Fn ( m u. { m } ) <-> ( f u. { <. m , U_ y e. ( f ` p ) pred ( y , A , R ) >. } ) Fn ( m u. { m } ) )
23	20, 22	sylibr 224	. . . 4 |- ( ( f Fn m /\ ( R FrSe A /\ ta ) ) -> G Fn ( m u. { m } ) )
24		fneq2 5980	. . . 4 |- ( n = ( m u. { m } ) -> ( G Fn n <-> G Fn ( m u. { m } ) ) )
25	23, 24	syl5ibr 236	. . 3 |- ( n = ( m u. { m } ) -> ( ( f Fn m /\ ( R FrSe A /\ ta ) ) -> G Fn n ) )
26	25	imp 445	. 2 |- ( ( n = ( m u. { m } ) /\ ( f Fn m /\ ( R FrSe A /\ ta ) ) ) -> G Fn n )
27	3, 26	sylbi 207	1 |- ( ( R FrSe A /\ ta /\ n = ( m u. { m } ) /\ f Fn m ) -> G Fn n )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   <.cop 4183   U_ciun 4520   dom cdm 5114   suc csuc 5725   Fun wfun 5882   Fn wfn 5883   ` cfv 5888   omcom 7065   /\ w-bnj17 30752   predc-bnj14 30754   FrSe w-bnj15 30758

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949  ax-reg 8497

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-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-bnj17 30753  df-bnj14 30755  df-bnj13 30757  df-bnj15 30759

This theorem is referenced by:  bnj543  30963