Theorem nfoi 8419

Description: Hypothesis builder for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
nfoi.1	|- F/_ x R
nfoi.2	|- F/_ x A

Assertion

Ref	Expression
nfoi	|- F/_ xOrdIso ( R , A )

Proof of Theorem nfoi

Dummy variables h a j t u v w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-oi 8415	. 2 |- OrdIso ( R , A ) = if ( ( R We A /\ R Se A ) , (recs ( ( h e. _V |-> ( iota_ v e. { w e. A | A. j e. ran h j R w } A. u e. { w e. A | A. j e. ran h j R w } -. u R v ) ) ) |` { a e. On | E. t e. A A. z e. (recs ( ( h e. _V |-> ( iota_ v e. { w e. A | A. j e. ran h j R w } A. u e. { w e. A | A. j e. ran h j R w } -. u R v ) ) ) " a ) z R t } ) , (/) )
2		nfoi.1	. . . . 5 |- F/_ x R
3		nfoi.2	. . . . 5 |- F/_ x A
4	2, 3	nfwe 5090	. . . 4 |- F/ x R We A
5	2, 3	nfse 5089	. . . 4 |- F/ x R Se A
6	4, 5	nfan 1828	. . 3 |- F/ x ( R We A /\ R Se A )
7		nfcv 2764	. . . . . 6 |- F/_ x _V
8		nfcv 2764	. . . . . . . . . 10 |- F/_ x ran h
9		nfcv 2764	. . . . . . . . . . 11 |- F/_ x j
10		nfcv 2764	. . . . . . . . . . 11 |- F/_ x w
11	9, 2, 10	nfbr 4699	. . . . . . . . . 10 |- F/ x j R w
12	8, 11	nfral 2945	. . . . . . . . 9 |- F/ x A. j e. ran h j R w
13	12, 3	nfrab 3123	. . . . . . . 8 |- F/_ x { w e. A | A. j e. ran h j R w }
14		nfcv 2764	. . . . . . . . . 10 |- F/_ x u
15		nfcv 2764	. . . . . . . . . 10 |- F/_ x v
16	14, 2, 15	nfbr 4699	. . . . . . . . 9 |- F/ x u R v
17	16	nfn 1784	. . . . . . . 8 |- F/ x -. u R v
18	13, 17	nfral 2945	. . . . . . 7 |- F/ x A. u e. { w e. A | A. j e. ran h j R w } -. u R v
19	18, 13	nfriota 6620	. . . . . 6 |- F/_ x ( iota_ v e. { w e. A | A. j e. ran h j R w } A. u e. { w e. A | A. j e. ran h j R w } -. u R v )
20	7, 19	nfmpt 4746	. . . . 5 |- F/_ x ( h e. _V |-> ( iota_ v e. { w e. A | A. j e. ran h j R w } A. u e. { w e. A | A. j e. ran h j R w } -. u R v ) )
21	20	nfrecs 7471	. . . 4 |- F/_ xrecs ( ( h e. _V |-> ( iota_ v e. { w e. A | A. j e. ran h j R w } A. u e. { w e. A | A. j e. ran h j R w } -. u R v ) ) )
22		nfcv 2764	. . . . . . . 8 |- F/_ x a
23	21, 22	nfima 5474	. . . . . . 7 |- F/_ x (recs ( ( h e. _V |-> ( iota_ v e. { w e. A | A. j e. ran h j R w } A. u e. { w e. A | A. j e. ran h j R w } -. u R v ) ) ) " a )
24		nfcv 2764	. . . . . . . 8 |- F/_ x z
25		nfcv 2764	. . . . . . . 8 |- F/_ x t
26	24, 2, 25	nfbr 4699	. . . . . . 7 |- F/ x z R t
27	23, 26	nfral 2945	. . . . . 6 |- F/ x A. z e. (recs ( ( h e. _V |-> ( iota_ v e. { w e. A | A. j e. ran h j R w } A. u e. { w e. A | A. j e. ran h j R w } -. u R v ) ) ) " a ) z R t
28	3, 27	nfrex 3007	. . . . 5 |- F/ x E. t e. A A. z e. (recs ( ( h e. _V |-> ( iota_ v e. { w e. A | A. j e. ran h j R w } A. u e. { w e. A | A. j e. ran h j R w } -. u R v ) ) ) " a ) z R t
29		nfcv 2764	. . . . 5 |- F/_ x On
30	28, 29	nfrab 3123	. . . 4 |- F/_ x { a e. On | E. t e. A A. z e. (recs ( ( h e. _V |-> ( iota_ v e. { w e. A | A. j e. ran h j R w } A. u e. { w e. A | A. j e. ran h j R w } -. u R v ) ) ) " a ) z R t }
31	21, 30	nfres 5398	. . 3 |- F/_ x (recs ( ( h e. _V |-> ( iota_ v e. { w e. A | A. j e. ran h j R w } A. u e. { w e. A | A. j e. ran h j R w } -. u R v ) ) ) |` { a e. On | E. t e. A A. z e. (recs ( ( h e. _V |-> ( iota_ v e. { w e. A | A. j e. ran h j R w } A. u e. { w e. A | A. j e. ran h j R w } -. u R v ) ) ) " a ) z R t } )
32		nfcv 2764	. . 3 |- F/_ x (/)
33	6, 31, 32	nfif 4115	. 2 |- F/_ x if ( ( R We A /\ R Se A ) , (recs ( ( h e. _V |-> ( iota_ v e. { w e. A | A. j e. ran h j R w } A. u e. { w e. A | A. j e. ran h j R w } -. u R v ) ) ) |` { a e. On | E. t e. A A. z e. (recs ( ( h e. _V |-> ( iota_ v e. { w e. A | A. j e. ran h j R w } A. u e. { w e. A | A. j e. ran h j R w } -. u R v ) ) ) " a ) z R t } ) , (/) )
34	1, 33	nfcxfr 2762	1 |- F/_ xOrdIso ( R , A )

Syntax hints:   -. wn 3   /\ wa 384   F/_wnfc 2751   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-3or 1038  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-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  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:  hsmexlem2  9249