Theorem bnj60 33072

Description: Well-founded recursion, part 1 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj60.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj60.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj60.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj60.4	|- F = U. C

Assertion

Ref	Expression
bnj60	|- ( R FrSe A -> F Fn A )

Distinct variable groups:   A, d, f, x   B, f   G, d, f, x   R, d, f, x

Allowed substitution hints:   B( x, d)   C( x, f, d)   F( x, f, d)   Y( x, f, d)

Proof of Theorem bnj60

Dummy variables g h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj60.1	. . . . 5 |- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
2		bnj60.2	. . . . 5 |- Y = <. x , ( f |` pred ( x , A , R ) ) >.
3		bnj60.3	. . . . 5 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4	1, 2, 3	bnj1497 33070	. . . 4 |- A. g e. C Fun g
5		eqid 2460	. . . . . . . 8 |- ( dom g i^i dom h ) = ( dom g i^i dom h )
6	1, 2, 3, 5	bnj1311 33034	. . . . . . 7 |- ( ( R FrSe A /\ g e. C /\ h e. C ) -> ( g |` ( dom g i^i dom h ) ) = ( h |` ( dom g i^i dom h ) ) )
7	6	3expia 1193	. . . . . 6 |- ( ( R FrSe A /\ g e. C ) -> ( h e. C -> ( g |` ( dom g i^i dom h ) ) = ( h |` ( dom g i^i dom h ) ) ) )
8	7	ralrimiv 2869	. . . . 5 |- ( ( R FrSe A /\ g e. C ) -> A. h e. C ( g |` ( dom g i^i dom h ) ) = ( h |` ( dom g i^i dom h ) ) )
9	8	ralrimiva 2871	. . . 4 |- ( R FrSe A -> A. g e. C A. h e. C ( g |` ( dom g i^i dom h ) ) = ( h |` ( dom g i^i dom h ) ) )
10		biid 236	. . . . 5 |- ( A. g e. C Fun g <-> A. g e. C Fun g )
11		biid 236	. . . . 5 |- ( ( A. g e. C Fun g /\ A. g e. C A. h e. C ( g |` ( dom g i^i dom h ) ) = ( h |` ( dom g i^i dom h ) ) ) <-> ( A. g e. C Fun g /\ A. g e. C A. h e. C ( g |` ( dom g i^i dom h ) ) = ( h |` ( dom g i^i dom h ) ) ) )
12	10, 5, 11	bnj1383 32844	. . . 4 |- ( ( A. g e. C Fun g /\ A. g e. C A. h e. C ( g |` ( dom g i^i dom h ) ) = ( h |` ( dom g i^i dom h ) ) ) -> Fun U. C )
13	4, 9, 12	sylancr 663	. . 3 |- ( R FrSe A -> Fun U. C )
14		bnj60.4	. . . 4 |- F = U. C
15	14	funeqi 5599	. . 3 |- ( Fun F <-> Fun U. C )
16	13, 15	sylibr 212	. 2 |- ( R FrSe A -> Fun F )
17	1, 2, 3, 14	bnj1498 33071	. 2 |- ( R FrSe A -> dom F = A )
18	16, 17	bnj1422 32850	1 |- ( R FrSe A -> F Fn A )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1374   e. wcel 1762   {cab 2445   A.wral 2807   E.wrex 2808   i^i cin 3468   C_ wss 3469   <.cop 4026   U.cuni 4238   dom cdm 4992   |` cres 4994   Fun wfun 5573   Fn wfn 5574   ` cfv 5579   predc-bnj14 32695   FrSe w-bnj15 32699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-reg 8007  ax-inf2 8047

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-om 6672  df-1o 7120  df-bnj17 32694  df-bnj14 32696  df-bnj13 32698  df-bnj15 32700  df-bnj18 32702  df-bnj19 32704

This theorem is referenced by:  bnj1501  33077  bnj1523  33081