Theorem finxpreclem3 33230

Description: Lemma for ^^ ^^ recursion theorems. (Contributed by ML, 20-Oct-2020.)

Hypothesis

Ref	Expression
finxpreclem3.1	|- 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
finxpreclem3	|- ( ( ( N e. om /\ 2o C_ N ) /\ X e. ( _V X. U ) ) -> <. U. N , ( 1st ` X ) >. = ( F ` <. N , X >. ) )

Distinct variable groups:   n, N, x   U, n, x   n, X, x

Allowed substitution hints:   F( x, n)

Proof of Theorem finxpreclem3

Step	Hyp	Ref	Expression
1		df-ov 6653	. 2 |- ( N F X ) = ( F ` <. N , X >. )
2		finxpreclem3.1	. . . 4 |- 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 >. ) ) )
3	2	a1i 11	. . 3 |- ( ( ( N e. om /\ 2o C_ N ) /\ X e. ( _V X. U ) ) -> 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 >. ) ) ) )
4		eqeq1 2626	. . . . . . 7 |- ( n = N -> ( n = 1o <-> N = 1o ) )
5		eleq1 2689	. . . . . . 7 |- ( x = X -> ( x e. U <-> X e. U ) )
6	4, 5	bi2anan9 917	. . . . . 6 |- ( ( n = N /\ x = X ) -> ( ( n = 1o /\ x e. U ) <-> ( N = 1o /\ X e. U ) ) )
7		eleq1 2689	. . . . . . . 8 |- ( x = X -> ( x e. ( _V X. U ) <-> X e. ( _V X. U ) ) )
8	7	adantl 482	. . . . . . 7 |- ( ( n = N /\ x = X ) -> ( x e. ( _V X. U ) <-> X e. ( _V X. U ) ) )
9		unieq 4444	. . . . . . . . 9 |- ( n = N -> U. n = U. N )
10	9	adantr 481	. . . . . . . 8 |- ( ( n = N /\ x = X ) -> U. n = U. N )
11		fveq2 6191	. . . . . . . . 9 |- ( x = X -> ( 1st ` x ) = ( 1st ` X ) )
12	11	adantl 482	. . . . . . . 8 |- ( ( n = N /\ x = X ) -> ( 1st ` x ) = ( 1st ` X ) )
13	10, 12	opeq12d 4410	. . . . . . 7 |- ( ( n = N /\ x = X ) -> <. U. n , ( 1st ` x ) >. = <. U. N , ( 1st ` X ) >. )
14		opeq12 4404	. . . . . . 7 |- ( ( n = N /\ x = X ) -> <. n , x >. = <. N , X >. )
15	8, 13, 14	ifbieq12d 4113	. . . . . 6 |- ( ( n = N /\ x = X ) -> if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) = if ( X e. ( _V X. U ) , <. U. N , ( 1st ` X ) >. , <. N , X >. ) )
16	6, 15	ifbieq2d 4111	. . . . 5 |- ( ( n = N /\ x = X ) -> if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) = if ( ( N = 1o /\ X e. U ) , (/) , if ( X e. ( _V X. U ) , <. U. N , ( 1st ` X ) >. , <. N , X >. ) ) )
17		sssucid 5802	. . . . . . . . . . . . 13 |- 1o C_ suc 1o
18		df-2o 7561	. . . . . . . . . . . . 13 |- 2o = suc 1o
19	17, 18	sseqtr4i 3638	. . . . . . . . . . . 12 |- 1o C_ 2o
20		1on 7567	. . . . . . . . . . . . . 14 |- 1o e. On
21	18, 20	sucneqoni 33214	. . . . . . . . . . . . 13 |- 2o =/= 1o
22	21	necomi 2848	. . . . . . . . . . . 12 |- 1o =/= 2o
23		df-pss 3590	. . . . . . . . . . . 12 |- ( 1o C. 2o <-> ( 1o C_ 2o /\ 1o =/= 2o ) )
24	19, 22, 23	mpbir2an 955	. . . . . . . . . . 11 |- 1o C. 2o
25		ssnpss 3710	. . . . . . . . . . 11 |- ( 2o C_ 1o -> -. 1o C. 2o )
26	24, 25	mt2 191	. . . . . . . . . 10 |- -. 2o C_ 1o
27		sseq2 3627	. . . . . . . . . 10 |- ( N = 1o -> ( 2o C_ N <-> 2o C_ 1o ) )
28	26, 27	mtbiri 317	. . . . . . . . 9 |- ( N = 1o -> -. 2o C_ N )
29	28	con2i 134	. . . . . . . 8 |- ( 2o C_ N -> -. N = 1o )
30	29	intnanrd 963	. . . . . . 7 |- ( 2o C_ N -> -. ( N = 1o /\ X e. U ) )
31	30	iffalsed 4097	. . . . . 6 |- ( 2o C_ 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 >. ) )
32		iftrue 4092	. . . . . 6 |- ( X e. ( _V X. U ) -> if ( X e. ( _V X. U ) , <. U. N , ( 1st ` X ) >. , <. N , X >. ) = <. U. N , ( 1st ` X ) >. )
33	31, 32	sylan9eq 2676	. . . . 5 |- ( ( 2o C_ N /\ X e. ( _V X. U ) ) -> if ( ( N = 1o /\ X e. U ) , (/) , if ( X e. ( _V X. U ) , <. U. N , ( 1st ` X ) >. , <. N , X >. ) ) = <. U. N , ( 1st ` X ) >. )
34	16, 33	sylan9eqr 2678	. . . 4 |- ( ( ( 2o C_ N /\ X e. ( _V X. U ) ) /\ ( n = N /\ x = X ) ) -> if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) = <. U. N , ( 1st ` X ) >. )
35	34	adantlll 754	. . 3 |- ( ( ( ( N e. om /\ 2o C_ N ) /\ X e. ( _V X. U ) ) /\ ( n = N /\ x = X ) ) -> if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) = <. U. N , ( 1st ` X ) >. )
36		simpll 790	. . 3 |- ( ( ( N e. om /\ 2o C_ N ) /\ X e. ( _V X. U ) ) -> N e. om )
37		elex 3212	. . . 4 |- ( X e. ( _V X. U ) -> X e. _V )
38	37	adantl 482	. . 3 |- ( ( ( N e. om /\ 2o C_ N ) /\ X e. ( _V X. U ) ) -> X e. _V )
39		opex 4932	. . . 4 |- <. U. N , ( 1st ` X ) >. e. _V
40	39	a1i 11	. . 3 |- ( ( ( N e. om /\ 2o C_ N ) /\ X e. ( _V X. U ) ) -> <. U. N , ( 1st ` X ) >. e. _V )
41	3, 35, 36, 38, 40	ovmpt2d 6788	. 2 |- ( ( ( N e. om /\ 2o C_ N ) /\ X e. ( _V X. U ) ) -> ( N F X ) = <. U. N , ( 1st ` X ) >. )
42	1, 41	syl5reqr 2671	1 |- ( ( ( N e. om /\ 2o C_ N ) /\ X e. ( _V X. U ) ) -> <. U. N , ( 1st ` X ) >. = ( F ` <. N , X >. ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   C_ wss 3574   C. wpss 3575   (/)c0 3915   ifcif 4086   <.cop 4183   U.cuni 4436   X. cxp 5112   suc csuc 5725   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   omcom 7065   1stc1st 7166   1oc1o 7553   2oc2o 7554

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  df-2o 7561

This theorem is referenced by:  finxpreclem4  33231