Theorem bnj60 33825

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 33823	. . . 4 |- A. g e. C Fun g
5		eqid 2441	. . . . . . . 8 |- ( dom g i^i dom h ) = ( dom g i^i dom h )
6	1, 2, 3, 5	bnj1311 33787	. . . . . . 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 1197	. . . . . 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 2853	. . . . 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 2855	. . . 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 33597	. . . 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 5594	. . 3 |- ( Fun F <-> Fun U. C )
16	13, 15	sylibr 212	. 2 |- ( R FrSe A -> Fun F )
17	1, 2, 3, 14	bnj1498 33824	. 2 |- ( R FrSe A -> dom F = A )
18	16, 17	bnj1422 33603	1 |- ( R FrSe A -> F Fn A )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1381   e. wcel 1802   {cab 2426   A.wral 2791   E.wrex 2792   i^i cin 3457   C_ wss 3458   <.cop 4016   U.cuni 4230   dom cdm 4985   |` cres 4987   Fun wfun 5568   Fn wfn 5569   ` cfv 5574   predc-bnj14 33448   FrSe w-bnj15 33452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-reg 8016  ax-inf2 8056

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-om 6682  df-1o 7128  df-bnj17 33447  df-bnj14 33449  df-bnj13 33451  df-bnj15 33453  df-bnj18 33455  df-bnj19 33457

This theorem is referenced by:  bnj1501  33830  bnj1523  33834