Theorem tfrcllemsucfn 6614

Description: We can extend an acceptable function by one element to produce a function. Lemma for tfrcl 6625. (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 2835	. . 3 |- ( ph -> z e. _V )
4		tfrcl.x	. . . . 5 |- ( ph -> Ord X )
5		ordelon 4523	. . . . 5 |- ( ( Ord X /\ z e. X ) -> z e. On )
6	4, 2, 5	syl2anc 415	. . . 4 |- ( ph -> z e. On )
7		eloni 4515	. . . 4 |- ( z e. On -> Ord z )
8		ordirr 4684	. . . 4 |- ( Ord z -> -. z e. z )
9	6, 7, 8	3syl 17	. . 3 |- ( ph -> -. z e. z )
10		feq2 5512	. . . . . . 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 1877	. . . . 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 1236	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( f : x --> S -> ( G ` f ) e. S ) )
15	14	alrimiv 1927	. . . . . 6 |- ( ( ph /\ x e. X ) -> A. f ( f : x --> S -> ( G ` f ) e. S ) )
16	15	ralrimiva 2623	. . . . 5 |- ( ph -> A. x e. X A. f ( f : x --> S -> ( G ` f ) e. S ) )
17	12, 16, 2	rspcdva 2934	. . . 4 |- ( ph -> A. f ( f : z --> S -> ( G ` f ) e. S ) )
18		feq1 5511	. . . . . 6 |- ( f = g -> ( f : z --> S <-> g : z --> S ) )
19		fveq2 5690	. . . . . . 7 |- ( f = g -> ( G ` f ) = ( G ` g ) )
20	19	eleq1d 2307	. . . . . 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 1913	. . . 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 5906	. . 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 1283	. 2 |- ( ph -> ( g u. { <. z , ( G ` g ) >. } ) : ( z u. { z } ) --> S )
26		df-suc 4511	. . 3 |- suc z = ( z u. { z } )
27	26	feq2i 5522	. 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 1009   A.wal 1400   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   E.wrex 2529   _Vcvv 2821   u. cun 3218   {csn 3705   <.cop 3708   Ord word 4502   Oncon0 4503   suc csuc 4505   |` cres 4771   Fun wfun 5366   -->wf 5368   ` cfv 5372  recscrecs 6565

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380

This theorem is referenced by:  tfrcllemsucaccv  6615  tfrcllembfn  6618