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Theorem frecabex 6302
Description: The class abstraction from df-frec 6295 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 4514 . . . 4 ω ∈ V
2 simpr 109 . . . . . . 7 ((dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚))) → 𝑥 ∈ (𝐹‘(𝑆𝑚)))
32abssi 3176 . . . . . 6 {𝑥 ∣ (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ⊆ (𝐹‘(𝑆𝑚))
4 frecabex.sex . . . . . . . 8 (𝜑𝑆𝑉)
5 vex 2692 . . . . . . . 8 𝑚 ∈ V
6 fvexg 5447 . . . . . . . 8 ((𝑆𝑉𝑚 ∈ V) → (𝑆𝑚) ∈ V)
74, 5, 6sylancl 410 . . . . . . 7 (𝜑 → (𝑆𝑚) ∈ V)
8 frecabex.fvex . . . . . . 7 (𝜑 → ∀𝑦(𝐹𝑦) ∈ V)
9 fveq2 5428 . . . . . . . . 9 (𝑦 = (𝑆𝑚) → (𝐹𝑦) = (𝐹‘(𝑆𝑚)))
109eleq1d 2209 . . . . . . . 8 (𝑦 = (𝑆𝑚) → ((𝐹𝑦) ∈ V ↔ (𝐹‘(𝑆𝑚)) ∈ V))
1110spcgv 2776 . . . . . . 7 ((𝑆𝑚) ∈ V → (∀𝑦(𝐹𝑦) ∈ V → (𝐹‘(𝑆𝑚)) ∈ V))
127, 8, 11sylc 62 . . . . . 6 (𝜑 → (𝐹‘(𝑆𝑚)) ∈ V)
13 ssexg 4074 . . . . . 6 (({𝑥 ∣ (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ⊆ (𝐹‘(𝑆𝑚)) ∧ (𝐹‘(𝑆𝑚)) ∈ V) → {𝑥 ∣ (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ∈ V)
143, 12, 13sylancr 411 . . . . 5 (𝜑 → {𝑥 ∣ (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ∈ V)
1514ralrimivw 2509 . . . 4 (𝜑 → ∀𝑚 ∈ ω {𝑥 ∣ (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ∈ V)
16 abrexex2g 6025 . . . 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 411 . . 3 (𝜑 → {𝑥 ∣ ∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ∈ V)
18 simpr 109 . . . . 5 ((dom 𝑆 = ∅ ∧ 𝑥𝐴) → 𝑥𝐴)
1918abssi 3176 . . . 4 {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥𝐴)} ⊆ 𝐴
20 frecabex.aex . . . 4 (𝜑𝐴𝑊)
21 ssexg 4074 . . . 4 (({𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥𝐴)} ⊆ 𝐴𝐴𝑊) → {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥𝐴)} ∈ V)
2219, 20, 21sylancr 411 . . 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 4370 . . 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 3347 . . . 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 2206 . . 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 698  wal 1330   = wceq 1332  wcel 1481  {cab 2126  wral 2417  wrex 2418  Vcvv 2689  cun 3073  wss 3075  c0 3367  suc csuc 4294  ωcom 4511  dom cdm 4546  cfv 5130
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-iinf 4509
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138
This theorem is referenced by:  frectfr  6304
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