Theorem wfr3 7435

Description: The principle of Well-Founded Recursion, part 3 of 3. Finally, we show that F is unique. We do this by showing that any function H with the same properties we proved of F in wfr1 7433 and wfr2 7434 is identical to F. (Contributed by Scott Fenton, 18-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)

Hypotheses

Ref	Expression
wfr3.1	|- R We A
wfr3.2	|- R Se A
wfr3.3	|- F = wrecs ( R , A , G )

Assertion

Ref	Expression
wfr3	|- ( ( H Fn A /\ A. z e. A ( H ` z ) = ( G ` ( H |` Pred ( R , A , z ) ) ) ) -> F = H )

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

Proof of Theorem wfr3

Step	Hyp	Ref	Expression
1		wfr3.1	. . 3 |- R We A
2		wfr3.2	. . 3 |- R Se A
3	1, 2	pm3.2i 471	. 2 |- ( R We A /\ R Se A )
4		wfr3.3	. . . 4 |- F = wrecs ( R , A , G )
5	1, 2, 4	wfr1 7433	. . 3 |- F Fn A
6	1, 2, 4	wfr2 7434	. . . 4 |- ( z e. A -> ( F ` z ) = ( G ` ( F |` Pred ( R , A , z ) ) ) )
7	6	rgen 2922	. . 3 |- A. z e. A ( F ` z ) = ( G ` ( F |` Pred ( R , A , z ) ) )
8	5, 7	pm3.2i 471	. 2 |- ( F Fn A /\ A. z e. A ( F ` z ) = ( G ` ( F |` Pred ( R , A , z ) ) ) )
9		wfr3g 7413	. 2 |- ( ( ( R We A /\ R Se A ) /\ ( F Fn A /\ A. z e. A ( F ` z ) = ( G ` ( F |` Pred ( R , A , z ) ) ) ) /\ ( H Fn A /\ A. z e. A ( H ` z ) = ( G ` ( H |` Pred ( R , A , z ) ) ) ) ) -> F = H )
10	3, 8, 9	mp3an12 1414	1 |- ( ( H Fn A /\ A. z e. A ( H ` z ) = ( G ` ( H |` Pred ( R , A , z ) ) ) ) -> F = H )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   A.wral 2912   Se wse 5071   We wwe 5072   |` cres 5116   Predcpred 5679   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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-wrecs 7407

This theorem is referenced by:  tfr3ALT  7498