Theorem heiborlem2 33611

Description: Lemma for heibor 33620. Substitutions for the set G. (Contributed by Jeff Madsen, 23-Jan-2014.)

Hypotheses

Ref	Expression
heibor.1	|- J = ( MetOpen ` D )
heibor.3	|- K = { u | -. E. v e. ( ~P U i^i Fin ) u C_ U. v }
heibor.4	|- G = { <. y , n >. | ( n e. NN0 /\ y e. ( F ` n ) /\ ( y B n ) e. K ) }
heiborlem2.5	|- A e. _V
heiborlem2.6	|- C e. _V

Assertion

Ref	Expression
heiborlem2	|- ( A G C <-> ( C e. NN0 /\ A e. ( F ` C ) /\ ( A B C ) e. K ) )

Distinct variable groups:   y, n, A   u, n, F, y   v, n, D, u, y   B, n, u, v, y   n, J, u, v, y   U, n, u, v, y   C, n, u, v, y   n, K, y

Allowed substitution hints:   A( v, u)   F( v)   G( y, v, u, n)   K( v, u)

Proof of Theorem heiborlem2

Step	Hyp	Ref	Expression
1		heiborlem2.5	. 2 |- A e. _V
2		heiborlem2.6	. 2 |- C e. _V
3		eleq1 2689	. . 3 |- ( y = A -> ( y e. ( F ` n ) <-> A e. ( F ` n ) ) )
4		oveq1 6657	. . . 4 |- ( y = A -> ( y B n ) = ( A B n ) )
5	4	eleq1d 2686	. . 3 |- ( y = A -> ( ( y B n ) e. K <-> ( A B n ) e. K ) )
6	3, 5	3anbi23d 1402	. 2 |- ( y = A -> ( ( n e. NN0 /\ y e. ( F ` n ) /\ ( y B n ) e. K ) <-> ( n e. NN0 /\ A e. ( F ` n ) /\ ( A B n ) e. K ) ) )
7		eleq1 2689	. . 3 |- ( n = C -> ( n e. NN0 <-> C e. NN0 ) )
8		fveq2 6191	. . . 4 |- ( n = C -> ( F ` n ) = ( F ` C ) )
9	8	eleq2d 2687	. . 3 |- ( n = C -> ( A e. ( F ` n ) <-> A e. ( F ` C ) ) )
10		oveq2 6658	. . . 4 |- ( n = C -> ( A B n ) = ( A B C ) )
11	10	eleq1d 2686	. . 3 |- ( n = C -> ( ( A B n ) e. K <-> ( A B C ) e. K ) )
12	7, 9, 11	3anbi123d 1399	. 2 |- ( n = C -> ( ( n e. NN0 /\ A e. ( F ` n ) /\ ( A B n ) e. K ) <-> ( C e. NN0 /\ A e. ( F ` C ) /\ ( A B C ) e. K ) ) )
13		heibor.4	. 2 |- G = { <. y , n >. | ( n e. NN0 /\ y e. ( F ` n ) /\ ( y B n ) e. K ) }
14	1, 2, 6, 12, 13	brab 4998	1 |- ( A G C <-> ( C e. NN0 /\ A e. ( F ` C ) /\ ( A B C ) e. K ) )

Syntax hints:   -. wn 3   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   class class class wbr 4653   {copab 4712   ` cfv 5888  (class class class)co 6650   Fincfn 7955   NN0cn0 11292   MetOpencmopn 19736

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  heiborlem3  33612  heiborlem5  33614  heiborlem6  33615  heiborlem8  33617  heiborlem10  33619