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Theorem frecabex 6263
Description: The class abstraction from df-frec 6256 exists. This is a lemma for other finite recursion proofs. (Contributed by Jim Kingdon, 13-May-2020.)
Hypotheses
Ref Expression
frecabex.sex (𝜑𝑆𝑉)
frecabex.fvex (𝜑 → ∀𝑦(𝐹𝑦) ∈ V)
frecabex.aex (𝜑𝐴𝑊)
Assertion
Ref Expression
frecabex (𝜑 → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚))) ∨ (dom 𝑆 = ∅ ∧ 𝑥𝐴))} ∈ V)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝑆,𝑦   𝜑,𝑚   𝑥,𝑚,𝑦   𝑦,𝐹
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑦,𝑚)   𝑆(𝑚)   𝐹(𝑚)   𝑉(𝑥,𝑦,𝑚)   𝑊(𝑥,𝑦,𝑚)

Proof of Theorem frecabex
StepHypRef Expression
1 omex 4477 . . . 4 ω ∈ V
2 simpr 109 . . . . . . 7 ((dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚))) → 𝑥 ∈ (𝐹‘(𝑆𝑚)))
32abssi 3142 . . . . . 6 {𝑥 ∣ (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ⊆ (𝐹‘(𝑆𝑚))
4 frecabex.sex . . . . . . . 8 (𝜑𝑆𝑉)
5 vex 2663 . . . . . . . 8 𝑚 ∈ V
6 fvexg 5408 . . . . . . . 8 ((𝑆𝑉𝑚 ∈ V) → (𝑆𝑚) ∈ V)
74, 5, 6sylancl 409 . . . . . . 7 (𝜑 → (𝑆𝑚) ∈ V)
8 frecabex.fvex . . . . . . 7 (𝜑 → ∀𝑦(𝐹𝑦) ∈ V)
9 fveq2 5389 . . . . . . . . 9 (𝑦 = (𝑆𝑚) → (𝐹𝑦) = (𝐹‘(𝑆𝑚)))
109eleq1d 2186 . . . . . . . 8 (𝑦 = (𝑆𝑚) → ((𝐹𝑦) ∈ V ↔ (𝐹‘(𝑆𝑚)) ∈ V))
1110spcgv 2747 . . . . . . 7 ((𝑆𝑚) ∈ V → (∀𝑦(𝐹𝑦) ∈ V → (𝐹‘(𝑆𝑚)) ∈ V))
127, 8, 11sylc 62 . . . . . 6 (𝜑 → (𝐹‘(𝑆𝑚)) ∈ V)
13 ssexg 4037 . . . . . 6 (({𝑥 ∣ (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ⊆ (𝐹‘(𝑆𝑚)) ∧ (𝐹‘(𝑆𝑚)) ∈ V) → {𝑥 ∣ (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ∈ V)
143, 12, 13sylancr 410 . . . . 5 (𝜑 → {𝑥 ∣ (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ∈ V)
1514ralrimivw 2483 . . . 4 (𝜑 → ∀𝑚 ∈ ω {𝑥 ∣ (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ∈ V)
16 abrexex2g 5986 . . . 4 ((ω ∈ V ∧ ∀𝑚 ∈ ω {𝑥 ∣ (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ∈ V) → {𝑥 ∣ ∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ∈ V)
171, 15, 16sylancr 410 . . 3 (𝜑 → {𝑥 ∣ ∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ∈ V)
18 simpr 109 . . . . 5 ((dom 𝑆 = ∅ ∧ 𝑥𝐴) → 𝑥𝐴)
1918abssi 3142 . . . 4 {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥𝐴)} ⊆ 𝐴
20 frecabex.aex . . . 4 (𝜑𝐴𝑊)
21 ssexg 4037 . . . 4 (({𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥𝐴)} ⊆ 𝐴𝐴𝑊) → {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥𝐴)} ∈ V)
2219, 20, 21sylancr 410 . . 3 (𝜑 → {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥𝐴)} ∈ V)
2317, 22jca 304 . 2 (𝜑 → ({𝑥 ∣ ∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ∈ V ∧ {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥𝐴)} ∈ V))
24 unexb 4333 . . 3 (({𝑥 ∣ ∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ∈ V ∧ {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥𝐴)} ∈ V) ↔ ({𝑥 ∣ ∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ∪ {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥𝐴)}) ∈ V)
25 unab 3313 . . . 4 ({𝑥 ∣ ∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ∪ {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥𝐴)}) = {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚))) ∨ (dom 𝑆 = ∅ ∧ 𝑥𝐴))}
2625eleq1i 2183 . . 3 (({𝑥 ∣ ∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ∪ {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥𝐴)}) ∈ V ↔ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚))) ∨ (dom 𝑆 = ∅ ∧ 𝑥𝐴))} ∈ V)
2724, 26bitri 183 . 2 (({𝑥 ∣ ∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ∈ V ∧ {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥𝐴)} ∈ V) ↔ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚))) ∨ (dom 𝑆 = ∅ ∧ 𝑥𝐴))} ∈ V)
2823, 27sylib 121 1 (𝜑 → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚))) ∨ (dom 𝑆 = ∅ ∧ 𝑥𝐴))} ∈ 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  cun 3039  wss 3041  c0 3333  suc csuc 4257  ωcom 4474  dom cdm 4509  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:  frectfr  6265
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