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Theorem frectfr 6401
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 6336 or tfri2d 6337, 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 2741 . . . . . . . 8 𝑔 ∈ V
21a1i 9 . . . . . . 7 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → 𝑔 ∈ V)
3 simpl 109 . . . . . . 7 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → ∀𝑧(𝐹𝑧) ∈ V)
4 simpr 110 . . . . . . 7 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → 𝐴𝑉)
52, 3, 4frecabex 6399 . . . . . 6 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))} ∈ V)
65ralrimivw 2551 . . . . 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 5343 . . . . 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 2741 . . . 4 𝑦 ∈ V
11 funfvex 5533 . . . . 5 ((Fun 𝐺𝑦 ∈ dom 𝐺) → (𝐺𝑦) ∈ V)
1211funfni 5317 . . . 4 ((𝐺 Fn V ∧ 𝑦 ∈ V) → (𝐺𝑦) ∈ V)
139, 10, 12sylancl 413 . . 3 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → (𝐺𝑦) ∈ V)
147funmpt2 5256 . . 3 Fun 𝐺
1513, 14jctil 312 . 2 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → (Fun 𝐺 ∧ (𝐺𝑦) ∈ V))
1615alrimiv 1874 1 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → ∀𝑦(Fun 𝐺 ∧ (𝐺𝑦) ∈ V))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wo 708  wal 1351   = wceq 1353  wcel 2148  {cab 2163  wral 2455  wrex 2456  Vcvv 2738  c0 3423  cmpt 4065  suc csuc 4366  ωcom 4590  dom cdm 4627  Fun wfun 5211   Fn wfn 5212  cfv 5217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225
This theorem is referenced by:  frecfnom  6402
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