Theorem tfrlemisucfn 6433

Description: We can extend an acceptable function by one element to produce a function. Lemma for tfrlemi1 6441. (Contributed by Jim Kingdon, 2-Jul-2019.)

Hypotheses

Ref	Expression
tfrlemisucfn.1	|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
tfrlemisucfn.2	|- ( ph -> A. x ( Fun F /\ ( F ` x ) e. _V ) )
tfrlemisucfn.3	|- ( ph -> z e. On )
tfrlemisucfn.4	|- ( ph -> g Fn z )
tfrlemisucfn.5	|- ( ph -> g e. A )

Assertion

Ref	Expression
tfrlemisucfn	|- ( ph -> ( g u. { <. z , ( F ` g ) >. } ) Fn suc z )

Distinct variable groups:   f, g, x, y, z, A   f, F, g, x, y, z   ph, y

Allowed substitution hints:   ph( x, z, f, g)

Proof of Theorem tfrlemisucfn

Step	Hyp	Ref	Expression
1		vex 2779	. . 3 |- z e. _V
2	1	a1i 9	. 2 |- ( ph -> z e. _V )
3		tfrlemisucfn.2	. . . 4 |- ( ph -> A. x ( Fun F /\ ( F ` x ) e. _V ) )
4	3	tfrlem3-2d 6421	. . 3 |- ( ph -> ( Fun F /\ ( F ` g ) e. _V ) )
5	4	simprd 114	. 2 |- ( ph -> ( F ` g ) e. _V )
6		tfrlemisucfn.4	. 2 |- ( ph -> g Fn z )
7		eqid 2207	. 2 |- ( g u. { <. z , ( F ` g ) >. } ) = ( g u. { <. z , ( F ` g ) >. } )
8		df-suc 4436	. 2 |- suc z = ( z u. { z } )
9		elirrv 4614	. . 3 |- -. z e. z
10	9	a1i 9	. 2 |- ( ph -> -. z e. z )
11	2, 5, 6, 7, 8, 10	fnunsn 5402	1 |- ( ph -> ( g u. { <. z , ( F ` g ) >. } ) Fn suc z )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   A.wal 1371   = wceq 1373   e. wcel 2178   {cab 2193   A.wral 2486   E.wrex 2487   _Vcvv 2776   u. cun 3172   {csn 3643   <.cop 3646   Oncon0 4428   suc csuc 4430   |` cres 4695   Fun wfun 5284   Fn wfn 5285   ` cfv 5290

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-suc 4436  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298

This theorem is referenced by:  tfrlemisucaccv  6434  tfrlemibfn  6437