Theorem wfrlem13 7427

Description: Lemma for well-founded recursion. From here through wfrlem16 7430, we aim to prove that dom F = A. We do this by supposing that there is an element z of A that is not in dom F. We then define C by extending dom F with the appropriate value at z. We then show that z cannot be an R minimal element of ( A \ dom F ), meaning that ( A \ dom F ) must be empty, so dom F = A. Here, we show that C is a function extending the domain of F by one. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)

Hypotheses

Ref	Expression
wfrlem13.1	|- R We A
wfrlem13.2	|- R Se A
wfrlem13.3	|- F = wrecs ( R , A , G )
wfrlem13.4	|- C = ( F u. { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } )

Assertion

Ref	Expression
wfrlem13	|- ( z e. ( A \ dom F ) -> C Fn ( dom F u. { z } ) )

Distinct variable groups:   z, A   z, F   z, R

Allowed substitution hints:   C( z)   G( z)

Proof of Theorem wfrlem13

Step	Hyp	Ref	Expression
1		wfrlem13.1	. . . . . 6 |- R We A
2		wfrlem13.2	. . . . . 6 |- R Se A
3		wfrlem13.3	. . . . . 6 |- F = wrecs ( R , A , G )
4	1, 2, 3	wfrfun 7425	. . . . 5 |- Fun F
5		vex 3203	. . . . . 6 |- z e. _V
6		fvex 6201	. . . . . 6 |- ( G ` ( F |` Pred ( R , A , z ) ) ) e. _V
7	5, 6	funsn 5939	. . . . 5 |- Fun { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. }
8	4, 7	pm3.2i 471	. . . 4 |- ( Fun F /\ Fun { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } )
9	6	dmsnop 5609	. . . . . 6 |- dom { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } = { z }
10	9	ineq2i 3811	. . . . 5 |- ( dom F i^i dom { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } ) = ( dom F i^i { z } )
11		eldifn 3733	. . . . . 6 |- ( z e. ( A \ dom F ) -> -. z e. dom F )
12		disjsn 4246	. . . . . 6 |- ( ( dom F i^i { z } ) = (/) <-> -. z e. dom F )
13	11, 12	sylibr 224	. . . . 5 |- ( z e. ( A \ dom F ) -> ( dom F i^i { z } ) = (/) )
14	10, 13	syl5eq 2668	. . . 4 |- ( z e. ( A \ dom F ) -> ( dom F i^i dom { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } ) = (/) )
15		funun 5932	. . . 4 |- ( ( ( Fun F /\ Fun { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } ) /\ ( dom F i^i dom { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } ) = (/) ) -> Fun ( F u. { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } ) )
16	8, 14, 15	sylancr 695	. . 3 |- ( z e. ( A \ dom F ) -> Fun ( F u. { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } ) )
17		dmun 5331	. . . 4 |- dom ( F u. { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } ) = ( dom F u. dom { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } )
18	9	uneq2i 3764	. . . 4 |- ( dom F u. dom { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } ) = ( dom F u. { z } )
19	17, 18	eqtri 2644	. . 3 |- dom ( F u. { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } ) = ( dom F u. { z } )
20	16, 19	jctir 561	. 2 |- ( z e. ( A \ dom F ) -> ( Fun ( F u. { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } ) /\ dom ( F u. { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } ) = ( dom F u. { z } ) ) )
21		wfrlem13.4	. . . 4 |- C = ( F u. { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } )
22	21	fneq1i 5985	. . 3 |- ( C Fn ( dom F u. { z } ) <-> ( F u. { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } ) Fn ( dom F u. { z } ) )
23		df-fn 5891	. . 3 |- ( ( F u. { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } ) Fn ( dom F u. { z } ) <-> ( Fun ( F u. { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } ) /\ dom ( F u. { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } ) = ( dom F u. { z } ) ) )
24	22, 23	bitri 264	. 2 |- ( C Fn ( dom F u. { z } ) <-> ( Fun ( F u. { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } ) /\ dom ( F u. { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } ) = ( dom F u. { z } ) ) )
25	20, 24	sylibr 224	1 |- ( z e. ( A \ dom F ) -> C Fn ( dom F u. { z } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   \ cdif 3571   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   <.cop 4183   Se wse 5071   We wwe 5072   dom cdm 5114   |` cres 5116   Predcpred 5679   Fun wfun 5882   Fn wfn 5883   ` cfv 5888  wrecscwrecs 7406

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-rmo 2920  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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  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-pred 5680  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-wrecs 7407

This theorem is referenced by:  wfrlem14  7428  wfrlem15  7429