Theorem tfr1onlemsucfn 6308

Description: We can extend an acceptable function by one element to produce a function. Lemma for tfr1on 6318. (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 2739	. 2 |- ( ph -> z e. _V )
3		fneq2 5277	. . . . . 6 |- ( x = z -> ( f Fn x <-> f Fn z ) )
4	3	imbi1d 230	. . . . 5 |- ( x = z -> ( ( f Fn x -> ( G ` f ) e. _V ) <-> ( f Fn z -> ( G ` f ) e. _V ) ) )
5	4	albidv 1812	. . . 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 1195	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( f Fn x -> ( G ` f ) e. _V ) )
8	7	alrimiv 1862	. . . . 5 |- ( ( ph /\ x e. X ) -> A. f ( f Fn x -> ( G ` f ) e. _V ) )
9	8	ralrimiva 2539	. . . 4 |- ( ph -> A. x e. X A. f ( f Fn x -> ( G ` f ) e. _V ) )
10	5, 9, 1	rspcdva 2835	. . 3 |- ( ph -> A. f ( f Fn z -> ( G ` f ) e. _V ) )
11		tfr1onlemsucfn.4	. . 3 |- ( ph -> g Fn z )
12		fneq1 5276	. . . . 5 |- ( f = g -> ( f Fn z <-> g Fn z ) )
13		fveq2 5486	. . . . . 6 |- ( f = g -> ( G ` f ) = ( G ` g ) )
14	13	eleq1d 2235	. . . . 5 |- ( f = g -> ( ( G ` f ) e. _V <-> ( G ` g ) e. _V ) )
15	12, 14	imbi12d 233	. . . 4 |- ( f = g -> ( ( f Fn z -> ( G ` f ) e. _V ) <-> ( g Fn z -> ( G ` g ) e. _V ) ) )
16	15	spv 1848	. . 3 |- ( A. f ( f Fn z -> ( G ` f ) e. _V ) -> ( g Fn z -> ( G ` g ) e. _V ) )
17	10, 11, 16	sylc 62	. 2 |- ( ph -> ( G ` g ) e. _V )
18		eqid 2165	. 2 |- ( g u. { <. z , ( G ` g ) >. } ) = ( g u. { <. z , ( G ` g ) >. } )
19		df-suc 4349	. 2 |- suc z = ( z u. { z } )
20		tfr1on.x	. . . 4 |- ( ph -> Ord X )
21		ordelon 4361	. . . 4 |- ( ( Ord X /\ z e. X ) -> z e. On )
22	20, 1, 21	syl2anc 409	. . 3 |- ( ph -> z e. On )
23		eloni 4353	. . 3 |- ( z e. On -> Ord z )
24		ordirr 4519	. . 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 5295	1 |- ( ph -> ( g u. { <. z , ( G ` g ) >. } ) Fn suc z )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   /\ w3a 968   A.wal 1341   = wceq 1343   e. wcel 2136   {cab 2151   A.wral 2444   E.wrex 2445   _Vcvv 2726   u. cun 3114   {csn 3576   <.cop 3579   Ord word 4340   Oncon0 4341   suc csuc 4343   |` cres 4606   Fun wfun 5182   Fn wfn 5183   ` cfv 5188  recscrecs 6272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fn 5191  df-fv 5196

This theorem is referenced by:  tfr1onlemsucaccv  6309  tfr1onlembfn  6312