Theorem tfrcllemsucfn 6597

Description: We can extend an acceptable function by one element to produce a function. Lemma for tfrcl 6608. (Contributed by Jim Kingdon, 24-Mar-2022.)

Hypotheses

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

Assertion

Ref	Expression
tfrcllemsucfn	|- ( ph -> ( g u. { <. z , ( G ` g ) >. } ) : suc z --> S )

Distinct variable groups:   f, G, x   S, f, 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)   S( y, z, g)   F( x, y, z, f, g)   G( y, z, g)   X( y, z, g)

Proof of Theorem tfrcllemsucfn

Step	Hyp	Ref	Expression
1		tfrcllemsucfn.4	. . 3 |- ( ph -> g : z --> S )
2		tfrcllemsucfn.3	. . . 4 |- ( ph -> z e. X )
3	2	elexd 2829	. . 3 |- ( ph -> z e. _V )
4		tfrcl.x	. . . . 5 |- ( ph -> Ord X )
5		ordelon 4509	. . . . 5 |- ( ( Ord X /\ z e. X ) -> z e. On )
6	4, 2, 5	syl2anc 411	. . . 4 |- ( ph -> z e. On )
7		eloni 4501	. . . 4 |- ( z e. On -> Ord z )
8		ordirr 4669	. . . 4 |- ( Ord z -> -. z e. z )
9	6, 7, 8	3syl 17	. . 3 |- ( ph -> -. z e. z )
10		feq2 5497	. . . . . . 7 |- ( x = z -> ( f : x --> S <-> f : z --> S ) )
11	10	imbi1d 231	. . . . . 6 |- ( x = z -> ( ( f : x --> S -> ( G ` f ) e. S ) <-> ( f : z --> S -> ( G ` f ) e. S ) ) )
12	11	albidv 1873	. . . . 5 |- ( x = z -> ( A. f ( f : x --> S -> ( G ` f ) e. S ) <-> A. f ( f : z --> S -> ( G ` f ) e. S ) ) )
13		tfrcl.ex	. . . . . . . 8 |- ( ( ph /\ x e. X /\ f : x --> S ) -> ( G ` f ) e. S )
14	13	3expia 1232	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( f : x --> S -> ( G ` f ) e. S ) )
15	14	alrimiv 1923	. . . . . 6 |- ( ( ph /\ x e. X ) -> A. f ( f : x --> S -> ( G ` f ) e. S ) )
16	15	ralrimiva 2617	. . . . 5 |- ( ph -> A. x e. X A. f ( f : x --> S -> ( G ` f ) e. S ) )
17	12, 16, 2	rspcdva 2928	. . . 4 |- ( ph -> A. f ( f : z --> S -> ( G ` f ) e. S ) )
18		feq1 5496	. . . . . 6 |- ( f = g -> ( f : z --> S <-> g : z --> S ) )
19		fveq2 5675	. . . . . . 7 |- ( f = g -> ( G ` f ) = ( G ` g ) )
20	19	eleq1d 2303	. . . . . 6 |- ( f = g -> ( ( G ` f ) e. S <-> ( G ` g ) e. S ) )
21	18, 20	imbi12d 234	. . . . 5 |- ( f = g -> ( ( f : z --> S -> ( G ` f ) e. S ) <-> ( g : z --> S -> ( G ` g ) e. S ) ) )
22	21	spv 1909	. . . 4 |- ( A. f ( f : z --> S -> ( G ` f ) e. S ) -> ( g : z --> S -> ( G ` g ) e. S ) )
23	17, 1, 22	sylc 62	. . 3 |- ( ph -> ( G ` g ) e. S )
24		fsnunf 5889	. . 3 |- ( ( g : z --> S /\ ( z e. _V /\ -. z e. z ) /\ ( G ` g ) e. S ) -> ( g u. { <. z , ( G ` g ) >. } ) : ( z u. { z } ) --> S )
25	1, 3, 9, 23, 24	syl121anc 1279	. 2 |- ( ph -> ( g u. { <. z , ( G ` g ) >. } ) : ( z u. { z } ) --> S )
26		df-suc 4497	. . 3 |- suc z = ( z u. { z } )
27	26	feq2i 5507	. 2 |- ( ( g u. { <. z , ( G ` g ) >. } ) : suc z --> S <-> ( g u. { <. z , ( G ` g ) >. } ) : ( z u. { z } ) --> S )
28	25, 27	sylibr 134	1 |- ( ph -> ( g u. { <. z , ( G ` g ) >. } ) : suc z --> S )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 1005   A.wal 1396   = wceq 1398   e. wcel 2205   {cab 2220   A.wral 2522   E.wrex 2523   _Vcvv 2815   u. cun 3212   {csn 3694   <.cop 3697   Ord word 4488   Oncon0 4489   suc csuc 4491   |` cres 4756   Fun wfun 5351   -->wf 5353   ` cfv 5357  recscrecs 6548

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365

This theorem is referenced by:  tfrcllemsucaccv  6598  tfrcllembfn  6601