Theorem ordtypelem1 8423

Description: Lemma for ordtype 8437. (Contributed by Mario Carneiro, 24-Jun-2015.)

Hypotheses

Ref	Expression
ordtypelem.1	|- F = recs ( G )
ordtypelem.2	|- C = { w e. A | A. j e. ran h j R w }
ordtypelem.3	|- G = ( h e. _V |-> ( iota_ v e. C A. u e. C -. u R v ) )
ordtypelem.5	|- T = { x e. On | E. t e. A A. z e. ( F " x ) z R t }
ordtypelem.6	|- O = OrdIso ( R , A )
ordtypelem.7	|- ( ph -> R We A )
ordtypelem.8	|- ( ph -> R Se A )

Assertion

Ref	Expression
ordtypelem1	|- ( ph -> O = ( F |` T ) )

Distinct variable groups:   v, u, C   h, j, t, u, v, w, x, z, R   A, h, j, t, u, v, w, x, z   t, O, u, v, x   ph, t, x   h, F, j, t, u, v, w, x, z

Allowed substitution hints:   ph( z, w, v, u, h, j)   C( x, z, w, t, h, j)   T( x, z, w, v, u, t, h, j)   G( x, z, w, v, u, t, h, j)   O( z, w, h, j)

Proof of Theorem ordtypelem1

Step	Hyp	Ref	Expression
1		ordtypelem.7	. . 3 |- ( ph -> R We A )
2		ordtypelem.8	. . 3 |- ( ph -> R Se A )
3		iftrue 4092	. . 3 |- ( ( R We A /\ R Se A ) -> if ( ( R We A /\ R Se A ) , ( F |` { x e. On | E. t e. A A. z e. ( F " x ) z R t } ) , (/) ) = ( F |` { x e. On | E. t e. A A. z e. ( F " x ) z R t } ) )
4	1, 2, 3	syl2anc 693	. 2 |- ( ph -> if ( ( R We A /\ R Se A ) , ( F |` { x e. On | E. t e. A A. z e. ( F " x ) z R t } ) , (/) ) = ( F |` { x e. On | E. t e. A A. z e. ( F " x ) z R t } ) )
5		ordtypelem.6	. . 3 |- O = OrdIso ( R , A )
6		ordtypelem.2	. . . 4 |- C = { w e. A | A. j e. ran h j R w }
7		ordtypelem.3	. . . 4 |- G = ( h e. _V |-> ( iota_ v e. C A. u e. C -. u R v ) )
8		ordtypelem.1	. . . 4 |- F = recs ( G )
9	6, 7, 8	dfoi 8416	. . 3 |- OrdIso ( R , A ) = if ( ( R We A /\ R Se A ) , ( F |` { x e. On | E. t e. A A. z e. ( F " x ) z R t } ) , (/) )
10	5, 9	eqtri 2644	. 2 |- O = if ( ( R We A /\ R Se A ) , ( F |` { x e. On | E. t e. A A. z e. ( F " x ) z R t } ) , (/) )
11		ordtypelem.5	. . 3 |- T = { x e. On | E. t e. A A. z e. ( F " x ) z R t }
12	11	reseq2i 5393	. 2 |- ( F |` T ) = ( F |` { x e. On | E. t e. A A. z e. ( F " x ) z R t } )
13	4, 10, 12	3eqtr4g 2681	1 |- ( ph -> O = ( F |` T ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   (/)c0 3915   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   Se wse 5071   We wwe 5072   ran crn 5115   |` cres 5116   "cima 5117   Oncon0 5723   iota_crio 6610  recscrecs 7467  OrdIsocoi 8414

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

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  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-mpt 4730  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-fv 5896  df-riota 6611  df-wrecs 7407  df-recs 7468  df-oi 8415

This theorem is referenced by:  ordtypelem4  8426  ordtypelem6  8428  ordtypelem7  8429  ordtypelem9  8431