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Theorem frrlem5e 29000
 Description: Lemma for founded recursion. The domain of the union of a subset of is closed under predecessors. (Contributed by Paul Chapman, 1-May-2012.)
Hypotheses
Ref Expression
frrlem5.1
frrlem5.2 Se
frrlem5.3
Assertion
Ref Expression
frrlem5e
Distinct variable groups:   ,,,   ,,,   ,,,   ,
Allowed substitution hints:   (,)   (,,)   (,,)

Proof of Theorem frrlem5e
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmuni 5212 . . . 4
21eleq2i 2545 . . 3
3 eliun 4330 . . 3
42, 3bitri 249 . 2
5 ssel2 3499 . . . . 5
6 frrlem5.3 . . . . . . . 8
76frrlem1 28992 . . . . . . 7
87abeq2i 2594 . . . . . 6
9 fndm 5680 . . . . . . . . 9
10 predeq3 28853 . . . . . . . . . . . . 13
1110sseq1d 3531 . . . . . . . . . . . 12
1211rspccv 3211 . . . . . . . . . . 11
13123ad2ant2 1018 . . . . . . . . . 10
14 eleq2 2540 . . . . . . . . . . 11
15 sseq2 3526 . . . . . . . . . . 11
1614, 15imbi12d 320 . . . . . . . . . 10
1713, 16syl5ibr 221 . . . . . . . . 9
189, 17syl 16 . . . . . . . 8
1918imp 429 . . . . . . 7
2019exlimiv 1698 . . . . . 6
218, 20sylbi 195 . . . . 5
225, 21syl 16 . . . 4
23 dmeq 5203 . . . . . . . . . 10
2423sseq2d 3532 . . . . . . . . 9
2524rspcev 3214 . . . . . . . 8
26 ssiun 4367 . . . . . . . 8
2725, 26syl 16 . . . . . . 7
28 dmuni 5212 . . . . . . 7
2927, 28syl6sseqr 3551 . . . . . 6
3029ex 434 . . . . 5
3130adantl 466 . . . 4
3222, 31syld 44 . . 3
3332rexlimdva 2955 . 2
344, 33syl5bi 217 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   w3a 973   wceq 1379  wex 1596   wcel 1767  cab 2452  wral 2814  wrex 2815   wss 3476  cuni 4245  ciun 4325   wfr 4835   Se wse 4836   cdm 4999   cres 5001   wfn 5583  cfv 5588  (class class class)co 6284  cpred 28848 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-fv 5596  df-ov 6287  df-pred 28849 This theorem is referenced by: (None)
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