Theorem tfrex 6577

Description: The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.)

Hypotheses

Ref	Expression
tfrex.1	|- F = recs ( G )
tfrex.2	|- ( ph -> A. x ( Fun G /\ ( G ` x ) e. _V ) )

Assertion

Ref	Expression
tfrex	|- ( ( ph /\ A e. V ) -> ( F ` A ) e. _V )

Distinct variable group:   x, G

Allowed substitution hints:   ph( x)   A( x)   F( x)   V( x)

Proof of Theorem tfrex

Dummy variables f g u y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tfrex.1	. . 3 |- F = recs ( G )
2	1	fveq1i 5649	. 2 |- ( F ` A ) = (recs ( G ) ` A )
3		eqid 2231	. . . 4 |- { g | E. z e. On ( g Fn z /\ A. u e. z ( g ` u ) = ( G ` ( g |` u ) ) ) } = { g | E. z e. On ( g Fn z /\ A. u e. z ( g ` u ) = ( G ` ( g |` u ) ) ) }
4	3	tfrlem3 6520	. . 3 |- { g | E. z e. On ( g Fn z /\ A. u e. z ( g ` u ) = ( G ` ( g |` u ) ) ) } = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }
5		tfrex.2	. . 3 |- ( ph -> A. x ( Fun G /\ ( G ` x ) e. _V ) )
6	4, 5	tfrexlem 6543	. 2 |- ( ( ph /\ A e. V ) -> (recs ( G ) ` A ) e. _V )
7	2, 6	eqeltrid 2318	1 |- ( ( ph /\ A e. V ) -> ( F ` A ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1396   = wceq 1398   e. wcel 2202   {cab 2217   A.wral 2511   E.wrex 2512   _Vcvv 2803   Oncon0 4466   |` cres 4733   Fun wfun 5327   Fn wfn 5328   ` cfv 5333  recscrecs 6513

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-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-recs 6514

This theorem is referenced by:  rdgexggg  6586