Theorem tfr1onlemsucfn 6277

Description: We can extend an acceptable function by one element to produce a function. Lemma for tfr1on 6287. (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 2722	. 2 |- ( ph -> z e. _V )
3		fneq2 5252	. . . . . 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 1801	. . . 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 1184	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( f Fn x -> ( G ` f ) e. _V ) )
8	7	alrimiv 1851	. . . . 5 |- ( ( ph /\ x e. X ) -> A. f ( f Fn x -> ( G ` f ) e. _V ) )
9	8	ralrimiva 2527	. . . 4 |- ( ph -> A. x e. X A. f ( f Fn x -> ( G ` f ) e. _V ) )
10	5, 9, 1	rspcdva 2818	. . 3 |- ( ph -> A. f ( f Fn z -> ( G ` f ) e. _V ) )
11		tfr1onlemsucfn.4	. . 3 |- ( ph -> g Fn z )
12		fneq1 5251	. . . . 5 |- ( f = g -> ( f Fn z <-> g Fn z ) )
13		fveq2 5461	. . . . . 6 |- ( f = g -> ( G ` f ) = ( G ` g ) )
14	13	eleq1d 2223	. . . . 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 1837	. . 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 2154	. 2 |- ( g u. { <. z , ( G ` g ) >. } ) = ( g u. { <. z , ( G ` g ) >. } )
19		df-suc 4326	. 2 |- suc z = ( z u. { z } )
20		tfr1on.x	. . . 4 |- ( ph -> Ord X )
21		ordelon 4338	. . . 4 |- ( ( Ord X /\ z e. X ) -> z e. On )
22	20, 1, 21	syl2anc 409	. . 3 |- ( ph -> z e. On )
23		eloni 4330	. . 3 |- ( z e. On -> Ord z )
24		ordirr 4495	. . 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 5270	1 |- ( ph -> ( g u. { <. z , ( G ` g ) >. } ) Fn suc z )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   /\ w3a 963   A.wal 1330   = wceq 1332   e. wcel 2125   {cab 2140   A.wral 2432   E.wrex 2433   _Vcvv 2709   u. cun 3096   {csn 3556   <.cop 3559   Ord word 4317   Oncon0 4318   suc csuc 4320   |` cres 4581   Fun wfun 5157   Fn wfn 5158   ` cfv 5163  recscrecs 6241

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-setind 4490

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-ral 2437  df-rex 2438  df-v 2711  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-nul 3391  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-br 3962  df-opab 4022  df-tr 4059  df-id 4248  df-iord 4321  df-on 4323  df-suc 4326  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-iota 5128  df-fun 5165  df-fn 5166  df-fv 5171

This theorem is referenced by:  tfr1onlemsucaccv  6278  tfr1onlembfn  6281