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Theorem tfrlem7 6180
 Description: Lemma for transfinite recursion. The union of all acceptable functions is a function. (Contributed by NM, 9-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.)
Hypothesis
Ref Expression
tfrlem.1
Assertion
Ref Expression
tfrlem7 recs
Distinct variable group:   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem tfrlem7
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . 3
21tfrlem6 6179 . 2 recs
31recsfval 6178 . . . . . . . . 9 recs
43eleq2i 2182 . . . . . . . 8 recs
5 eluni 3707 . . . . . . . 8
64, 5bitri 183 . . . . . . 7 recs
73eleq2i 2182 . . . . . . . 8 recs
8 eluni 3707 . . . . . . . 8
97, 8bitri 183 . . . . . . 7 recs
106, 9anbi12i 453 . . . . . 6 recs recs
11 eeanv 1882 . . . . . 6
1210, 11bitr4i 186 . . . . 5 recs recs
13 df-br 3898 . . . . . . . . 9
14 df-br 3898 . . . . . . . . 9
1513, 14anbi12i 453 . . . . . . . 8
161tfrlem5 6177 . . . . . . . . 9
1716impcom 124 . . . . . . . 8
1815, 17sylanbr 281 . . . . . . 7
1918an4s 560 . . . . . 6
2019exlimivv 1850 . . . . 5
2112, 20sylbi 120 . . . 4 recs recs
2221ax-gen 1408 . . 3 recs recs
2322gen2 1409 . 2 recs recs
24 dffun4 5102 . 2 recs recs recs recs
252, 23, 24mpbir2an 909 1 recs
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103  wal 1312   wceq 1314  wex 1451   wcel 1463  cab 2101  wral 2391  wrex 2392  cop 3498  cuni 3704   class class class wbr 3897  con0 4253   cres 4509   wrel 4512   wfun 5085   wfn 5086  cfv 5091  recscrecs 6167 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-setind 4420 This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-res 4519  df-iota 5056  df-fun 5093  df-fn 5094  df-fv 5099  df-recs 6168 This theorem is referenced by:  tfrlem9  6182  tfrfun  6183  tfrlemibfn  6191  tfrlemiubacc  6193  tfri1d  6198  rdgfun  6236
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