Theorem bnj60 31130

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 31128	. . . 4 |- A. g e. C Fun g
5		eqid 2622	. . . . . . . 8 |- ( dom g i^i dom h ) = ( dom g i^i dom h )
6	1, 2, 3, 5	bnj1311 31092	. . . . . . 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 1267	. . . . . 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 2965	. . . . 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 2966	. . . 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 251	. . . . 5 |- ( A. g e. C Fun g <-> A. g e. C Fun g )
11		biid 251	. . . . 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 30902	. . . 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 695	. . 3 |- ( R FrSe A -> Fun U. C )
14		bnj60.4	. . . 4 |- F = U. C
15	14	funeqi 5909	. . 3 |- ( Fun F <-> Fun U. C )
16	13, 15	sylibr 224	. 2 |- ( R FrSe A -> Fun F )
17	1, 2, 3, 14	bnj1498 31129	. 2 |- ( R FrSe A -> dom F = A )
18	16, 17	bnj1422 30908	1 |- ( R FrSe A -> F Fn A )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   <.cop 4183   U.cuni 4436   dom cdm 5114   |` cres 5116   Fun wfun 5882   Fn wfn 5883   ` cfv 5888   predc-bnj14 30754   FrSe w-bnj15 30758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-bnj17 30753  df-bnj14 30755  df-bnj13 30757  df-bnj15 30759  df-bnj18 30761  df-bnj19 30763

This theorem is referenced by:  bnj1501  31135  bnj1523  31139