Theorem tfrlem6 7105

Description: Lemma for transfinite recursion. The union of all acceptable functions is a relation. (Contributed by NM, 8-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)

Hypothesis

Ref	Expression
tfrlem.1	|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }

Assertion

Ref	Expression
tfrlem6	|- Rel recs ( F )

Distinct variable group:   x, f, y, F

Allowed substitution hints:   A( x, y, f)

Proof of Theorem tfrlem6

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reluni 4959	. . 3 |- ( Rel U. A <-> A. g e. A Rel g )
2		tfrlem.1	. . . . 5 |- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
3	2	tfrlem4 7102	. . . 4 |- ( g e. A -> Fun g )
4		funrel 5602	. . . 4 |- ( Fun g -> Rel g )
5	3, 4	syl 17	. . 3 |- ( g e. A -> Rel g )
6	1, 5	mprgbir 2754	. 2 |- Rel U. A
7	2	recsfval 7104	. . 3 |- recs ( F ) = U. A
8	7	releqi 4921	. 2 |- ( Rel recs ( F ) <-> Rel U. A )
9	6, 8	mpbir 213	1 |- Rel recs ( F )

Syntax hints:   /\ wa 371   = wceq 1446   e. wcel 1889   {cab 2439   A.wral 2739   E.wrex 2740   U.cuni 4201   |` cres 4839   Rel wrel 4842   Oncon0 5426   Fun wfun 5579   Fn wfn 5580   ` cfv 5585  recscrecs 7094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642  ax-un 6588

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-tr 4501  df-eprel 4748  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-iota 5549  df-fun 5587  df-fn 5588  df-fv 5593  df-wrecs 7033  df-recs 7095

This theorem is referenced by:  tfrlem7  7106  tfrlem11  7111  tfrlem15  7115  tfrlem16  7116