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Theorem frectfr 6015
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 5979 or tfri2d 5980, 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 2577 . . . . . . . 8 𝑔 ∈ V
21a1i 9 . . . . . . 7 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → 𝑔 ∈ V)
3 simpl 106 . . . . . . 7 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → ∀𝑧(𝐹𝑧) ∈ V)
4 simpr 107 . . . . . . 7 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → 𝐴𝑉)
52, 3, 4frecabex 6014 . . . . . 6 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))} ∈ V)
65ralrimivw 2410 . . . . 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 5052 . . . . 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 2577 . . . 4 𝑦 ∈ V
11 funfvex 5219 . . . . 5 ((Fun 𝐺𝑦 ∈ dom 𝐺) → (𝐺𝑦) ∈ V)
1211funfni 5026 . . . 4 ((𝐺 Fn V ∧ 𝑦 ∈ V) → (𝐺𝑦) ∈ V)
139, 10, 12sylancl 398 . . 3 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → (𝐺𝑦) ∈ V)
147funmpt2 4966 . . 3 Fun 𝐺
1513, 14jctil 299 . 2 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → (Fun 𝐺 ∧ (𝐺𝑦) ∈ V))
1615alrimiv 1770 1 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → ∀𝑦(Fun 𝐺 ∧ (𝐺𝑦) ∈ V))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wo 639  wal 1257   = wceq 1259  wcel 1409  {cab 2042  wral 2323  wrex 2324  Vcvv 2574  c0 3251  cmpt 3845  suc csuc 4129  ωcom 4340  dom cdm 4372  Fun wfun 4923   Fn wfn 4924  cfv 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937
This theorem is referenced by:  frecfnom  6016  frecsuclem1  6017  frecsuclemdm  6018  frecsuclem3  6020
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