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Theorem frecabex 5984
 Description: The class abstraction from df-frec 5978 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 4316 . . . 4 ω ∈ V
2 simpr 103 . . . . . . 7 ((dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚))) → 𝑥 ∈ (𝐹‘(𝑆𝑚)))
32abssi 3015 . . . . . 6 {𝑥 ∣ (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ⊆ (𝐹‘(𝑆𝑚))
4 frecabex.sex . . . . . . . 8 (𝜑𝑆𝑉)
5 vex 2560 . . . . . . . 8 𝑚 ∈ V
6 fvexg 5194 . . . . . . . 8 ((𝑆𝑉𝑚 ∈ V) → (𝑆𝑚) ∈ V)
74, 5, 6sylancl 392 . . . . . . 7 (𝜑 → (𝑆𝑚) ∈ V)
8 frecabex.fvex . . . . . . 7 (𝜑 → ∀𝑦(𝐹𝑦) ∈ V)
9 fveq2 5178 . . . . . . . . 9 (𝑦 = (𝑆𝑚) → (𝐹𝑦) = (𝐹‘(𝑆𝑚)))
109eleq1d 2106 . . . . . . . 8 (𝑦 = (𝑆𝑚) → ((𝐹𝑦) ∈ V ↔ (𝐹‘(𝑆𝑚)) ∈ V))
1110spcgv 2640 . . . . . . 7 ((𝑆𝑚) ∈ V → (∀𝑦(𝐹𝑦) ∈ V → (𝐹‘(𝑆𝑚)) ∈ V))
127, 8, 11sylc 56 . . . . . 6 (𝜑 → (𝐹‘(𝑆𝑚)) ∈ V)
13 ssexg 3896 . . . . . 6 (({𝑥 ∣ (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ⊆ (𝐹‘(𝑆𝑚)) ∧ (𝐹‘(𝑆𝑚)) ∈ V) → {𝑥 ∣ (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ∈ V)
143, 12, 13sylancr 393 . . . . 5 (𝜑 → {𝑥 ∣ (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ∈ V)
1514ralrimivw 2393 . . . 4 (𝜑 → ∀𝑚 ∈ ω {𝑥 ∣ (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ∈ V)
16 abrexex2g 5747 . . . 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 393 . . 3 (𝜑 → {𝑥 ∣ ∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ∈ V)
18 simpr 103 . . . . 5 ((dom 𝑆 = ∅ ∧ 𝑥𝐴) → 𝑥𝐴)
1918abssi 3015 . . . 4 {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥𝐴)} ⊆ 𝐴
20 frecabex.aex . . . 4 (𝜑𝐴𝑊)
21 ssexg 3896 . . . 4 (({𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥𝐴)} ⊆ 𝐴𝐴𝑊) → {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥𝐴)} ∈ V)
2219, 20, 21sylancr 393 . . 3 (𝜑 → {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥𝐴)} ∈ V)
2317, 22jca 290 . 2 (𝜑 → ({𝑥 ∣ ∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚)))} ∈ V ∧ {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥𝐴)} ∈ V))
24 unexb 4177 . . 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 3204 . . . 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 2103 . . 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 173 . 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 127 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 97   ∨ wo 629  ∀wal 1241   = wceq 1243   ∈ wcel 1393  {cab 2026  ∀wral 2306  ∃wrex 2307  Vcvv 2557   ∪ cun 2915   ⊆ wss 2917  ∅c0 3224  suc csuc 4102  ωcom 4313  dom cdm 4345  ‘cfv 4902 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-iinf 4311 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910 This theorem is referenced by:  frectfr  5985  frecsuclem3  5990
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