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Theorem rdg0 8033
 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 8029 . . . 4 Lim dom rec(𝐹, 𝐴)
2 limomss 7561 . . . 4 (Lim dom rec(𝐹, 𝐴) → ω ⊆ dom rec(𝐹, 𝐴))
31, 2ax-mp 5 . . 3 ω ⊆ dom rec(𝐹, 𝐴)
4 peano1 7577 . . 3 ∅ ∈ ω
53, 4sselii 3940 . 2 ∅ ∈ dom rec(𝐹, 𝐴)
6 eqid 2820 . . 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 8031 . . 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 8018 . 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 1537   ∈ wcel 2114  Vcvv 3473   ⊆ wss 3912  ∅c0 4267  ifcif 4441  ∪ cuni 4812   ↦ cmpt 5120  dom cdm 5529  ran crn 5530  Lim wlim 6166  ‘cfv 6329  ωcom 7556  reccrdg 8021 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5177  ax-nul 5184  ax-pow 5240  ax-pr 5304  ax-un 7437 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3752  df-csb 3860  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-pss 3930  df-nul 4268  df-if 4442  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4813  df-iun 4895  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5434  df-eprel 5439  df-po 5448  df-so 5449  df-fr 5488  df-we 5490  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-pred 6122  df-ord 6168  df-on 6169  df-lim 6170  df-suc 6171  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-fv 6337  df-om 7557  df-wrecs 7923  df-recs 7984  df-rdg 8022 This theorem is referenced by:  rdg0g  8039  seqomlem1  8062  seqomlem3  8064  om0  8118  oe0  8123  oev2  8124  r10  9173  aleph0  9468  ackbij2lem2  9638  ackbij2lem3  9639  satfv0  32610  satf00  32626  rdgprc  33044  finxp0  34686  finxp1o  34687  finxpreclem4  34689  finxpreclem6  34691
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