Theorem tfr1onlemsucfn 6398

Description: We can extend an acceptable function by one element to produce a function. Lemma for tfr1on 6408. (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 2776	. 2 |- ( ph -> z e. _V )
3		fneq2 5347	. . . . . 6 |- ( x = z -> ( f Fn x <-> f Fn z ) )
4	3	imbi1d 231	. . . . 5 |- ( x = z -> ( ( f Fn x -> ( G ` f ) e. _V ) <-> ( f Fn z -> ( G ` f ) e. _V ) ) )
5	4	albidv 1838	. . . 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 1207	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( f Fn x -> ( G ` f ) e. _V ) )
8	7	alrimiv 1888	. . . . 5 |- ( ( ph /\ x e. X ) -> A. f ( f Fn x -> ( G ` f ) e. _V ) )
9	8	ralrimiva 2570	. . . 4 |- ( ph -> A. x e. X A. f ( f Fn x -> ( G ` f ) e. _V ) )
10	5, 9, 1	rspcdva 2873	. . 3 |- ( ph -> A. f ( f Fn z -> ( G ` f ) e. _V ) )
11		tfr1onlemsucfn.4	. . 3 |- ( ph -> g Fn z )
12		fneq1 5346	. . . . 5 |- ( f = g -> ( f Fn z <-> g Fn z ) )
13		fveq2 5558	. . . . . 6 |- ( f = g -> ( G ` f ) = ( G ` g ) )
14	13	eleq1d 2265	. . . . 5 |- ( f = g -> ( ( G ` f ) e. _V <-> ( G ` g ) e. _V ) )
15	12, 14	imbi12d 234	. . . 4 |- ( f = g -> ( ( f Fn z -> ( G ` f ) e. _V ) <-> ( g Fn z -> ( G ` g ) e. _V ) ) )
16	15	spv 1874	. . 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 2196	. 2 |- ( g u. { <. z , ( G ` g ) >. } ) = ( g u. { <. z , ( G ` g ) >. } )
19		df-suc 4406	. 2 |- suc z = ( z u. { z } )
20		tfr1on.x	. . . 4 |- ( ph -> Ord X )
21		ordelon 4418	. . . 4 |- ( ( Ord X /\ z e. X ) -> z e. On )
22	20, 1, 21	syl2anc 411	. . 3 |- ( ph -> z e. On )
23		eloni 4410	. . 3 |- ( z e. On -> Ord z )
24		ordirr 4578	. . 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 5365	1 |- ( ph -> ( g u. { <. z , ( G ` g ) >. } ) Fn suc z )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 980   A.wal 1362   = wceq 1364   e. wcel 2167   {cab 2182   A.wral 2475   E.wrex 2476   _Vcvv 2763   u. cun 3155   {csn 3622   <.cop 3625   Ord word 4397   Oncon0 4398   suc csuc 4400   |` cres 4665   Fun wfun 5252   Fn wfn 5253   ` cfv 5258  recscrecs 6362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266

This theorem is referenced by:  tfr1onlemsucaccv  6399  tfr1onlembfn  6402