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Theorem frecabex 6119
Description: The class abstraction from df-frec 6112 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 4383 . . . 4 ω ∈ V
2 simpr 108 . . . . . . 7 ((dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚))) → 𝑥 ∈ (𝐹‘(𝑆𝑚)))
32abssi 3085 . . . . . 6 {𝑥 ∣ (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ⊆ (𝐹‘(𝑆𝑚))
4 frecabex.sex . . . . . . . 8 (𝜑𝑆𝑉)
5 vex 2618 . . . . . . . 8 𝑚 ∈ V
6 fvexg 5289 . . . . . . . 8 ((𝑆𝑉𝑚 ∈ V) → (𝑆𝑚) ∈ V)
74, 5, 6sylancl 404 . . . . . . 7 (𝜑 → (𝑆𝑚) ∈ V)
8 frecabex.fvex . . . . . . 7 (𝜑 → ∀𝑦(𝐹𝑦) ∈ V)
9 fveq2 5270 . . . . . . . . 9 (𝑦 = (𝑆𝑚) → (𝐹𝑦) = (𝐹‘(𝑆𝑚)))
109eleq1d 2153 . . . . . . . 8 (𝑦 = (𝑆𝑚) → ((𝐹𝑦) ∈ V ↔ (𝐹‘(𝑆𝑚)) ∈ V))
1110spcgv 2699 . . . . . . 7 ((𝑆𝑚) ∈ V → (∀𝑦(𝐹𝑦) ∈ V → (𝐹‘(𝑆𝑚)) ∈ V))
127, 8, 11sylc 61 . . . . . 6 (𝜑 → (𝐹‘(𝑆𝑚)) ∈ V)
13 ssexg 3955 . . . . . 6 (({𝑥 ∣ (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ⊆ (𝐹‘(𝑆𝑚)) ∧ (𝐹‘(𝑆𝑚)) ∈ V) → {𝑥 ∣ (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ∈ V)
143, 12, 13sylancr 405 . . . . 5 (𝜑 → {𝑥 ∣ (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ∈ V)
1514ralrimivw 2443 . . . 4 (𝜑 → ∀𝑚 ∈ ω {𝑥 ∣ (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ∈ V)
16 abrexex2g 5850 . . . 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 405 . . 3 (𝜑 → {𝑥 ∣ ∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ∈ V)
18 simpr 108 . . . . 5 ((dom 𝑆 = ∅ ∧ 𝑥𝐴) → 𝑥𝐴)
1918abssi 3085 . . . 4 {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥𝐴)} ⊆ 𝐴
20 frecabex.aex . . . 4 (𝜑𝐴𝑊)
21 ssexg 3955 . . . 4 (({𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥𝐴)} ⊆ 𝐴𝐴𝑊) → {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥𝐴)} ∈ V)
2219, 20, 21sylancr 405 . . 3 (𝜑 → {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥𝐴)} ∈ V)
2317, 22jca 300 . 2 (𝜑 → ({𝑥 ∣ ∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ∈ V ∧ {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥𝐴)} ∈ V))
24 unexb 4243 . . 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 3255 . . . 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 2150 . . 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 182 . 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 120 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 102  wo 662  wal 1285   = wceq 1287  wcel 1436  {cab 2071  wral 2355  wrex 2356  Vcvv 2615  cun 2986  wss 2988  c0 3275  suc csuc 4168  ωcom 4380  dom cdm 4413  cfv 4983
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3931  ax-sep 3934  ax-pow 3986  ax-pr 4012  ax-un 4236  ax-iinf 4378
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-int 3674  df-iun 3717  df-br 3823  df-opab 3877  df-mpt 3878  df-id 4096  df-iom 4381  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426  df-iota 4948  df-fun 4985  df-fn 4986  df-f 4987  df-f1 4988  df-fo 4989  df-f1o 4990  df-fv 4991
This theorem is referenced by:  frectfr  6121
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