Theorem bnj60 32355

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 32353	. . . 4 |- A. g e. C Fun g
5		eqid 2451	. . . . . . . 8 |- ( dom g i^i dom h ) = ( dom g i^i dom h )
6	1, 2, 3, 5	bnj1311 32317	. . . . . . 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 1190	. . . . . 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 2820	. . . . 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 2822	. . . 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 32127	. . . 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 5538	. . 3 |- ( Fun F <-> Fun U. C )
16	13, 15	sylibr 212	. 2 |- ( R FrSe A -> Fun F )
17	1, 2, 3, 14	bnj1498 32354	. 2 |- ( R FrSe A -> dom F = A )
18	16, 17	bnj1422 32133	1 |- ( R FrSe A -> F Fn A )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1370   e. wcel 1758   {cab 2436   A.wral 2795   E.wrex 2796   i^i cin 3427   C_ wss 3428   <.cop 3983   U.cuni 4191   dom cdm 4940   |` cres 4942   Fun wfun 5512   Fn wfn 5513   ` cfv 5518   predc-bnj14 31978   FrSe w-bnj15 31982

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-reg 7910  ax-inf2 7950

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-om 6579  df-1o 7022  df-bnj17 31977  df-bnj14 31979  df-bnj13 31981  df-bnj15 31983  df-bnj18 31985  df-bnj19 31987

This theorem is referenced by:  bnj1501  32360  bnj1523  32364