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Theorem frectfr 6265
Description: Lemma to connect transfinite recursion theorems with finite recursion. That is, given the conditions 𝐹 Fn V and 𝐴𝑉 on frec(𝐹, 𝐴), we want to be able to apply tfri1d 6200 or tfri2d 6201, and this lemma lets us satisfy hypotheses of those theorems.

(Contributed by Jim Kingdon, 15-Aug-2019.)

Hypothesis
Ref Expression
frectfr.1 𝐺 = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})
Assertion
Ref Expression
frectfr ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → ∀𝑦(Fun 𝐺 ∧ (𝐺𝑦) ∈ V))
Distinct variable groups:   𝑔,𝑚,𝑥,𝑦,𝐴   𝑧,𝑔,𝐹,𝑚,𝑥,𝑦   𝑔,𝑉,𝑚,𝑦
Allowed substitution hints:   𝐴(𝑧)   𝐺(𝑥,𝑦,𝑧,𝑔,𝑚)   𝑉(𝑥,𝑧)

Proof of Theorem frectfr
StepHypRef Expression
1 vex 2663 . . . . . . . 8 𝑔 ∈ V
21a1i 9 . . . . . . 7 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → 𝑔 ∈ V)
3 simpl 108 . . . . . . 7 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → ∀𝑧(𝐹𝑧) ∈ V)
4 simpr 109 . . . . . . 7 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → 𝐴𝑉)
52, 3, 4frecabex 6263 . . . . . 6 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))} ∈ V)
65ralrimivw 2483 . . . . 5 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → ∀𝑔 ∈ V {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))} ∈ V)
7 frectfr.1 . . . . . 6 𝐺 = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})
87fnmpt 5219 . . . . 5 (∀𝑔 ∈ V {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))} ∈ V → 𝐺 Fn V)
96, 8syl 14 . . . 4 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → 𝐺 Fn V)
10 vex 2663 . . . 4 𝑦 ∈ V
11 funfvex 5406 . . . . 5 ((Fun 𝐺𝑦 ∈ dom 𝐺) → (𝐺𝑦) ∈ V)
1211funfni 5193 . . . 4 ((𝐺 Fn V ∧ 𝑦 ∈ V) → (𝐺𝑦) ∈ V)
139, 10, 12sylancl 409 . . 3 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → (𝐺𝑦) ∈ V)
147funmpt2 5132 . . 3 Fun 𝐺
1513, 14jctil 310 . 2 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → (Fun 𝐺 ∧ (𝐺𝑦) ∈ V))
1615alrimiv 1830 1 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → ∀𝑦(Fun 𝐺 ∧ (𝐺𝑦) ∈ V))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wo 682  wal 1314   = wceq 1316  wcel 1465  {cab 2103  wral 2393  wrex 2394  Vcvv 2660  c0 3333  cmpt 3959  suc csuc 4257  ωcom 4474  dom cdm 4509  Fun wfun 5087   Fn wfn 5088  cfv 5093
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101
This theorem is referenced by:  frecfnom  6266
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