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Theorem wfr2 25487
 Description: The Principle of Well-Founded Recursion, part 2 of 3. Next, we show that the value of at any is recursively applied to all "previous" values of . (Contributed by Scott Fenton, 18-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
wfr2.1
wfr2.2 Se
wfr2.3
wfr2.4
Assertion
Ref Expression
wfr2
Distinct variable groups:   ,,,   ,,,   ,,,   ,,,   ,
Allowed substitution hints:   ()   (,,,)   ()   ()   ()

Proof of Theorem wfr2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fveq2 5687 . . 3
2 predeq3 25385 . . . . 5
32reseq2d 5105 . . . 4
43fveq2d 5691 . . 3
51, 4eqeq12d 2418 . 2
6 wfr2.1 . . . . 5
7 wfr2.2 . . . . 5 Se
8 wfr2.3 . . . . 5
9 wfr2.4 . . . . 5
10 eqid 2404 . . . . 5
116, 7, 8, 9, 10wfrlem16 25485 . . . 4
1211eleq2i 2468 . . 3
136, 7, 8, 9wfrlem12 25481 . . 3
1412, 13sylbir 205 . 2
155, 14vtoclga 2977 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936  wex 1547   wceq 1649   wcel 1721  cab 2390  wral 2666   cun 3278   wss 3280  csn 3774  cop 3777  cuni 3975   Se wse 4499   wwe 4500   cdm 4837   cres 4839   wfn 5408  cfv 5413  cpred 25381 This theorem is referenced by:  wfr2c  25488  wfr3  25489  tfr2ALT  25492 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-pred 25382
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