Theorem brfi1uzindOLD 13286

Description: Obsolete version of brfi1uzind 13280 as of 28-Mar-2021. (Contributed by AV, 7-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
brfi1uzindOLD.r	|- Rel G
brfi1uzindOLD.f	|- F e. U
brfi1uzindOLD.l	|- L e. NN0
brfi1uzindOLD.1	|- ( ( v = V /\ e = E ) -> ( ps <-> ph ) )
brfi1uzindOLD.2	|- ( ( v = w /\ e = f ) -> ( ps <-> th ) )
brfi1uzindOLD.3	|- ( ( v G e /\ n e. v ) -> ( v \ { n } ) G F )
brfi1uzindOLD.4	|- ( ( w = ( v \ { n } ) /\ f = F ) -> ( th <-> ch ) )
brfi1uzindOLD.base	|- ( ( v G e /\ ( # ` v ) = L ) -> ps )
brfi1uzindOLD.step	|- ( ( ( ( y + 1 ) e. NN0 /\ ( v G e /\ ( # ` v ) = ( y + 1 ) /\ n e. v ) ) /\ ch ) -> ps )

Assertion

Ref	Expression
brfi1uzindOLD	|- ( ( V G E /\ V e. Fin /\ L <_ ( # ` V ) ) -> ph )

Distinct variable groups:   e, E, n, v   f, F, w   e, G, f, n, v, w, y   e, L, n, v, y   e, V, n, v   ps, f, n, w, y   th, e, n, v   ch, f, w   ph, e, n, v

Allowed substitution hints:   ph( y, w, f)   ps( v, e)   ch( y, v, e, n)   th( y, w, f)   U( y, w, v, e, f, n)   E( y, w, f)   F( y, v, e, n)   L( w, f)   V( y, w, f)

Proof of Theorem brfi1uzindOLD

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brfi1uzindOLD.r	. . . 4 |- Rel G
2		brrelex12 5155	. . . 4 |- ( ( Rel G /\ V G E ) -> ( V e. _V /\ E e. _V ) )
3	1, 2	mpan 706	. . 3 |- ( V G E -> ( V e. _V /\ E e. _V ) )
4		simpl 473	. . . . 5 |- ( ( V e. _V /\ E e. _V ) -> V e. _V )
5		simplr 792	. . . . . 6 |- ( ( ( V e. _V /\ E e. _V ) /\ a = V ) -> E e. _V )
6		breq12 4658	. . . . . . 7 |- ( ( a = V /\ b = E ) -> ( a G b <-> V G E ) )
7	6	adantll 750	. . . . . 6 |- ( ( ( ( V e. _V /\ E e. _V ) /\ a = V ) /\ b = E ) -> ( a G b <-> V G E ) )
8	5, 7	sbcied 3472	. . . . 5 |- ( ( ( V e. _V /\ E e. _V ) /\ a = V ) -> ( [. E / b ]. a G b <-> V G E ) )
9	4, 8	sbcied 3472	. . . 4 |- ( ( V e. _V /\ E e. _V ) -> ( [. V / a ]. [. E / b ]. a G b <-> V G E ) )
10	9	biimprcd 240	. . 3 |- ( V G E -> ( ( V e. _V /\ E e. _V ) -> [. V / a ]. [. E / b ]. a G b ) )
11	3, 10	mpd 15	. 2 |- ( V G E -> [. V / a ]. [. E / b ]. a G b )
12		brfi1uzindOLD.f	. . 3 |- F e. U
13		brfi1uzindOLD.l	. . 3 |- L e. NN0
14		brfi1uzindOLD.1	. . 3 |- ( ( v = V /\ e = E ) -> ( ps <-> ph ) )
15		brfi1uzindOLD.2	. . 3 |- ( ( v = w /\ e = f ) -> ( ps <-> th ) )
16		vex 3203	. . . . 5 |- v e. _V
17		vex 3203	. . . . 5 |- e e. _V
18		breq12 4658	. . . . 5 |- ( ( a = v /\ b = e ) -> ( a G b <-> v G e ) )
19	16, 17, 18	sbc2ie 3505	. . . 4 |- ( [. v / a ]. [. e / b ]. a G b <-> v G e )
20		brfi1uzindOLD.3	. . . . 5 |- ( ( v G e /\ n e. v ) -> ( v \ { n } ) G F )
21		difexg 4808	. . . . . . 7 |- ( v e. _V -> ( v \ { n } ) e. _V )
22	16, 21	ax-mp 5	. . . . . 6 |- ( v \ { n } ) e. _V
23	12	elexi 3213	. . . . . 6 |- F e. _V
24		breq12 4658	. . . . . 6 |- ( ( a = ( v \ { n } ) /\ b = F ) -> ( a G b <-> ( v \ { n } ) G F ) )
25	22, 23, 24	sbc2ie 3505	. . . . 5 |- ( [. ( v \ { n } ) / a ]. [. F / b ]. a G b <-> ( v \ { n } ) G F )
26	20, 25	sylibr 224	. . . 4 |- ( ( v G e /\ n e. v ) -> [. ( v \ { n } ) / a ]. [. F / b ]. a G b )
27	19, 26	sylanb 489	. . 3 |- ( ( [. v / a ]. [. e / b ]. a G b /\ n e. v ) -> [. ( v \ { n } ) / a ]. [. F / b ]. a G b )
28		brfi1uzindOLD.4	. . 3 |- ( ( w = ( v \ { n } ) /\ f = F ) -> ( th <-> ch ) )
29		brfi1uzindOLD.base	. . . 4 |- ( ( v G e /\ ( # ` v ) = L ) -> ps )
30	19, 29	sylanb 489	. . 3 |- ( ( [. v / a ]. [. e / b ]. a G b /\ ( # ` v ) = L ) -> ps )
31	19	3anbi1i 1253	. . . . 5 |- ( ( [. v / a ]. [. e / b ]. a G b /\ ( # ` v ) = ( y + 1 ) /\ n e. v ) <-> ( v G e /\ ( # ` v ) = ( y + 1 ) /\ n e. v ) )
32	31	anbi2i 730	. . . 4 |- ( ( ( y + 1 ) e. NN0 /\ ( [. v / a ]. [. e / b ]. a G b /\ ( # ` v ) = ( y + 1 ) /\ n e. v ) ) <-> ( ( y + 1 ) e. NN0 /\ ( v G e /\ ( # ` v ) = ( y + 1 ) /\ n e. v ) ) )
33		brfi1uzindOLD.step	. . . 4 |- ( ( ( ( y + 1 ) e. NN0 /\ ( v G e /\ ( # ` v ) = ( y + 1 ) /\ n e. v ) ) /\ ch ) -> ps )
34	32, 33	sylanb 489	. . 3 |- ( ( ( ( y + 1 ) e. NN0 /\ ( [. v / a ]. [. e / b ]. a G b /\ ( # ` v ) = ( y + 1 ) /\ n e. v ) ) /\ ch ) -> ps )
35	12, 13, 14, 15, 27, 28, 30, 34	fi1uzindOLD 13285	. 2 |- ( ( [. V / a ]. [. E / b ]. a G b /\ V e. Fin /\ L <_ ( # ` V ) ) -> ph )
36	11, 35	syl3an1 1359	1 |- ( ( V G E /\ V e. Fin /\ L <_ ( # ` V ) ) -> ph )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   [.wsbc 3435   \ cdif 3571   {csn 4177   class class class wbr 4653   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   Fincfn 7955   1c1 9937   + caddc 9939   <_ cle 10075   NN0cn0 11292   #chash 13117

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-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  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-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118

This theorem is referenced by:  brfi1indOLD  13287