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Theorem tfrlem9a 7073
 Description: Lemma for transfinite recursion. Without using ax-rep 4568, show that all the restrictions of recs are sets. (Contributed by Mario Carneiro, 16-Nov-2014.)
Hypothesis
Ref Expression
tfrlem.1
Assertion
Ref Expression
tfrlem9a recs recs
Distinct variable groups:   ,,,   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem tfrlem9a
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . . . 5
21tfrlem7 7070 . . . 4 recs
3 funfvop 6000 . . . 4 recs recs recs recs
42, 3mpan 670 . . 3 recs recs recs
51recsfval 7068 . . . . 5 recs
65eleq2i 2535 . . . 4 recs recs recs
7 eluni 4254 . . . 4 recs recs
86, 7bitri 249 . . 3 recs recs recs
94, 8sylib 196 . 2 recs recs
10 simprr 757 . . . 4 recs recs
11 vex 3112 . . . . 5
121, 11tfrlem3a 7064 . . . 4
1310, 12sylib 196 . . 3 recs recs
142a1i 11 . . . . . . . 8 recs recs recs
15 simplrr 762 . . . . . . . . . 10 recs recs
16 elssuni 4281 . . . . . . . . . 10
1715, 16syl 16 . . . . . . . . 9 recs recs
1817, 5syl6sseqr 3546 . . . . . . . 8 recs recs recs
19 fndm 5686 . . . . . . . . . . . 12
2019ad2antll 728 . . . . . . . . . . 11 recs recs
21 simprl 756 . . . . . . . . . . 11 recs recs
2220, 21eqeltrd 2545 . . . . . . . . . 10 recs recs
23 eloni 4897 . . . . . . . . . 10
2422, 23syl 16 . . . . . . . . 9 recs recs
25 simpll 753 . . . . . . . . . 10 recs recs recs
26 fvex 5882 . . . . . . . . . . 11 recs
2726a1i 11 . . . . . . . . . 10 recs recs recs
28 simplrl 761 . . . . . . . . . . 11 recs recs recs
29 df-br 4457 . . . . . . . . . . 11 recs recs
3028, 29sylibr 212 . . . . . . . . . 10 recs recs recs
31 breldmg 5218 . . . . . . . . . 10 recs recs recs
3225, 27, 30, 31syl3anc 1228 . . . . . . . . 9 recs recs
33 ordelss 4903 . . . . . . . . 9
3424, 32, 33syl2anc 661 . . . . . . . 8 recs recs
35 fun2ssres 5635 . . . . . . . 8 recs recs recs
3614, 18, 34, 35syl3anc 1228 . . . . . . 7 recs recs recs
3711resex 5327 . . . . . . . 8
3837a1i 11 . . . . . . 7 recs recs
3936, 38eqeltrd 2545 . . . . . 6 recs recs recs
4039expr 615 . . . . 5 recs recs recs
4140adantrd 468 . . . 4 recs recs recs
4241rexlimdva 2949 . . 3 recs recs recs
4313, 42mpd 15 . 2 recs recs recs
449, 43exlimddv 1727 1 recs recs
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1395  wex 1613   wcel 1819  cab 2442  wral 2807  wrex 2808  cvv 3109   wss 3471  cop 4038  cuni 4251   class class class wbr 4456   word 4886  con0 4887   cdm 5008   cres 5010   wfun 5588   wfn 5589  cfv 5594  recscrecs 7059 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-recs 7060 This theorem is referenced by:  tfrlem15  7079  tfrlem16  7080  rdgseg  7106
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