Theorem finxpreclem5 33232

Description: Lemma for ^^ ^^ recursion theorems. (Contributed by ML, 24-Oct-2020.)

Hypothesis

Ref	Expression
finxpreclem5.1	|- F = ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) )

Assertion

Ref	Expression
finxpreclem5	|- ( ( n e. om /\ 1o e. n ) -> ( -. x e. ( _V X. U ) -> ( F ` <. n , x >. ) = <. n , x >. ) )

Distinct variable group:   x, n

Allowed substitution hints:   U( x, n)   F( x, n)

Proof of Theorem finxpreclem5

Step	Hyp	Ref	Expression
1		df-ov 6653	. . 3 |- ( n F x ) = ( F ` <. n , x >. )
2		vex 3203	. . . . . 6 |- x e. _V
3		0ex 4790	. . . . . . 7 |- (/) e. _V
4		opex 4932	. . . . . . . 8 |- <. U. n , ( 1st ` x ) >. e. _V
5		opex 4932	. . . . . . . 8 |- <. n , x >. e. _V
6	4, 5	ifex 4156	. . . . . . 7 |- if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) e. _V
7	3, 6	ifex 4156	. . . . . 6 |- if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) e. _V
8		finxpreclem5.1	. . . . . . 7 |- F = ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) )
9	8	ovmpt4g 6783	. . . . . 6 |- ( ( n e. om /\ x e. _V /\ if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) e. _V ) -> ( n F x ) = if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) )
10	2, 7, 9	mp3an23 1416	. . . . 5 |- ( n e. om -> ( n F x ) = if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) )
11	10	ad2antrr 762	. . . 4 |- ( ( ( n e. om /\ 1o e. n ) /\ -. x e. ( _V X. U ) ) -> ( n F x ) = if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) )
12		1on 7567	. . . . . . . . . . 11 |- 1o e. On
13	12	onirri 5834	. . . . . . . . . 10 |- -. 1o e. 1o
14		eleq2 2690	. . . . . . . . . 10 |- ( n = 1o -> ( 1o e. n <-> 1o e. 1o ) )
15	13, 14	mtbiri 317	. . . . . . . . 9 |- ( n = 1o -> -. 1o e. n )
16	15	con2i 134	. . . . . . . 8 |- ( 1o e. n -> -. n = 1o )
17	16	intnanrd 963	. . . . . . 7 |- ( 1o e. n -> -. ( n = 1o /\ x e. U ) )
18	17	iffalsed 4097	. . . . . 6 |- ( 1o e. n -> if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) = if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) )
19	18	adantl 482	. . . . 5 |- ( ( n e. om /\ 1o e. n ) -> if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) = if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) )
20		iffalse 4095	. . . . 5 |- ( -. x e. ( _V X. U ) -> if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) = <. n , x >. )
21	19, 20	sylan9eq 2676	. . . 4 |- ( ( ( n e. om /\ 1o e. n ) /\ -. x e. ( _V X. U ) ) -> if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) = <. n , x >. )
22	11, 21	eqtrd 2656	. . 3 |- ( ( ( n e. om /\ 1o e. n ) /\ -. x e. ( _V X. U ) ) -> ( n F x ) = <. n , x >. )
23	1, 22	syl5eqr 2670	. 2 |- ( ( ( n e. om /\ 1o e. n ) /\ -. x e. ( _V X. U ) ) -> ( F ` <. n , x >. ) = <. n , x >. )
24	23	ex 450	1 |- ( ( n e. om /\ 1o e. n ) -> ( -. x e. ( _V X. U ) -> ( F ` <. n , x >. ) = <. n , x >. ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   ifcif 4086   <.cop 4183   U.cuni 4436   X. cxp 5112   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   omcom 7065   1stc1st 7166   1oc1o 7553

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-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

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-rab 2921  df-v 3202  df-sbc 3436  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-br 4654  df-opab 4713  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-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1o 7560

This theorem is referenced by:  finxpreclem6  33233