Theorem tfr1onlemsucfn 6037

Description: We can extend an acceptable function by one element to produce a function. Lemma for tfr1on 6047. (Contributed by Jim Kingdon, 12-Mar-2022.)

Hypotheses

Ref	Expression
tfr1on.f	|- F = recs ( G )
tfr1on.g	|- ( ph -> Fun G )
tfr1on.x	|- ( ph -> Ord X )
tfr1on.ex	|- ( ( ph /\ x e. X /\ f Fn x ) -> ( G ` f ) e. _V )
tfr1onlemsucfn.1	|- A = { f | E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }
tfr1onlemsucfn.3	|- ( ph -> z e. X )
tfr1onlemsucfn.4	|- ( ph -> g Fn z )
tfr1onlemsucfn.5	|- ( ph -> g e. A )

Assertion

Ref	Expression
tfr1onlemsucfn	|- ( ph -> ( g u. { <. z , ( G ` g ) >. } ) Fn suc z )

Distinct variable groups:   f, G, x   f, X, x   f, g   ph, f, x   z, f, x

Allowed substitution hints:   ph( y, z, g)   A( x, y, z, f, g)   F( x, y, z, f, g)   G( y, z, g)   X( y, z, g)

Proof of Theorem tfr1onlemsucfn

Step	Hyp	Ref	Expression
1		tfr1onlemsucfn.3	. . 3 |- ( ph -> z e. X )
2	1	elexd 2623	. 2 |- ( ph -> z e. _V )
3		fneq2 5056	. . . . . 6 |- ( x = z -> ( f Fn x <-> f Fn z ) )
4	3	imbi1d 229	. . . . 5 |- ( x = z -> ( ( f Fn x -> ( G ` f ) e. _V ) <-> ( f Fn z -> ( G ` f ) e. _V ) ) )
5	4	albidv 1747	. . . 4 |- ( x = z -> ( A. f ( f Fn x -> ( G ` f ) e. _V ) <-> A. f ( f Fn z -> ( G ` f ) e. _V ) ) )
6		tfr1on.ex	. . . . . . 7 |- ( ( ph /\ x e. X /\ f Fn x ) -> ( G ` f ) e. _V )
7	6	3expia 1141	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( f Fn x -> ( G ` f ) e. _V ) )
8	7	alrimiv 1797	. . . . 5 |- ( ( ph /\ x e. X ) -> A. f ( f Fn x -> ( G ` f ) e. _V ) )
9	8	ralrimiva 2440	. . . 4 |- ( ph -> A. x e. X A. f ( f Fn x -> ( G ` f ) e. _V ) )
10	5, 9, 1	rspcdva 2717	. . 3 |- ( ph -> A. f ( f Fn z -> ( G ` f ) e. _V ) )
11		tfr1onlemsucfn.4	. . 3 |- ( ph -> g Fn z )
12		fneq1 5055	. . . . 5 |- ( f = g -> ( f Fn z <-> g Fn z ) )
13		fveq2 5253	. . . . . 6 |- ( f = g -> ( G ` f ) = ( G ` g ) )
14	13	eleq1d 2151	. . . . 5 |- ( f = g -> ( ( G ` f ) e. _V <-> ( G ` g ) e. _V ) )
15	12, 14	imbi12d 232	. . . 4 |- ( f = g -> ( ( f Fn z -> ( G ` f ) e. _V ) <-> ( g Fn z -> ( G ` g ) e. _V ) ) )
16	15	spv 1783	. . 3 |- ( A. f ( f Fn z -> ( G ` f ) e. _V ) -> ( g Fn z -> ( G ` g ) e. _V ) )
17	10, 11, 16	sylc 61	. 2 |- ( ph -> ( G ` g ) e. _V )
18		eqid 2083	. 2 |- ( g u. { <. z , ( G ` g ) >. } ) = ( g u. { <. z , ( G ` g ) >. } )
19		df-suc 4162	. 2 |- suc z = ( z u. { z } )
20		tfr1on.x	. . . 4 |- ( ph -> Ord X )
21		ordelon 4174	. . . 4 |- ( ( Ord X /\ z e. X ) -> z e. On )
22	20, 1, 21	syl2anc 403	. . 3 |- ( ph -> z e. On )
23		eloni 4166	. . 3 |- ( z e. On -> Ord z )
24		ordirr 4321	. . 3 |- ( Ord z -> -. z e. z )
25	22, 23, 24	3syl 17	. 2 |- ( ph -> -. z e. z )
26	2, 17, 11, 18, 19, 25	fnunsn 5074	1 |- ( ph -> ( g u. { <. z , ( G ` g ) >. } ) Fn suc z )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   /\ w3a 920   A.wal 1283   = wceq 1285   e. wcel 1434   {cab 2069   A.wral 2353   E.wrex 2354   _Vcvv 2612   u. cun 2982   {csn 3422   <.cop 3425   Ord word 4153   Oncon0 4154   suc csuc 4156   |` cres 4403   Fun wfun 4963   Fn wfn 4964   ` cfv 4969  recscrecs 6001

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000  ax-setind 4316

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-tr 3902  df-id 4084  df-iord 4157  df-on 4159  df-suc 4162  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-iota 4934  df-fun 4971  df-fn 4972  df-fv 4977

This theorem is referenced by:  tfr1onlemsucaccv  6038  tfr1onlembfn  6041