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Theorem rdg0 7562
Description: The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
Hypothesis
Ref Expression
rdg.1 𝐴 ∈ V
Assertion
Ref Expression
rdg0 (rec(𝐹, 𝐴)‘∅) = 𝐴

Proof of Theorem rdg0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rdgdmlim 7558 . . . 4 Lim dom rec(𝐹, 𝐴)
2 limomss 7112 . . . 4 (Lim dom rec(𝐹, 𝐴) → ω ⊆ dom rec(𝐹, 𝐴))
31, 2ax-mp 5 . . 3 ω ⊆ dom rec(𝐹, 𝐴)
4 peano1 7127 . . 3 ∅ ∈ ω
53, 4sselii 3633 . 2 ∅ ∈ dom rec(𝐹, 𝐴)
6 eqid 2651 . . 3 (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐹‘(𝑥 dom 𝑥))))) = (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐹‘(𝑥 dom 𝑥)))))
7 rdgvalg 7560 . . 3 (𝑦 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴)‘𝑦) = ((𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐹‘(𝑥 dom 𝑥)))))‘(rec(𝐹, 𝐴) ↾ 𝑦)))
8 rdg.1 . . 3 𝐴 ∈ V
96, 7, 8tz7.44-1 7547 . 2 (∅ ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴)‘∅) = 𝐴)
105, 9ax-mp 5 1 (rec(𝐹, 𝐴)‘∅) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wcel 2030  Vcvv 3231  wss 3607  c0 3948  ifcif 4119   cuni 4468  cmpt 4762  dom cdm 5143  ran crn 5144  Lim wlim 5762  cfv 5926  ωcom 7107  reccrdg 7550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551
This theorem is referenced by:  rdg0g  7568  seqomlem1  7590  seqomlem3  7592  om0  7642  oe0  7647  oev2  7648  r10  8669  aleph0  8927  ackbij2lem2  9100  ackbij2lem3  9101  rdgprc  31824  finxp0  33358  finxp1o  33359  finxpreclem4  33361  finxpreclem6  33363
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