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Theorem frectfr 6394
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 6329 or tfri2d 6330, 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 2740 . . . . . . . 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 6392 . . . . . 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 5337 . . . . 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 2740 . . . 4 𝑦 ∈ V
11 funfvex 5527 . . . . 5 ((Fun 𝐺𝑦 ∈ dom 𝐺) → (𝐺𝑦) ∈ V)
1211funfni 5311 . . . 4 ((𝐺 Fn V ∧ 𝑦 ∈ V) → (𝐺𝑦) ∈ V)
139, 10, 12sylancl 413 . . 3 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → (𝐺𝑦) ∈ V)
147funmpt2 5250 . . 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 2737  c0 3422  cmpt 4061  suc csuc 4361  ωcom 4585  dom cdm 4622  Fun wfun 5205   Fn wfn 5206  cfv 5211
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 4115  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-iinf 4583
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 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219
This theorem is referenced by:  frecfnom  6395
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