Theorem frecabex 6629

Description: The class abstraction from df-frec 6622 exists. This is a lemma for other finite recursion proofs. (Contributed by Jim Kingdon, 13-May-2020.)

Hypotheses

Ref	Expression
frecabex.sex	⊢ (𝜑 → 𝑆 ∈ 𝑉)
frecabex.fvex	⊢ (𝜑 → ∀𝑦(𝐹‘𝑦) ∈ V)
frecabex.aex	⊢ (𝜑 → 𝐴 ∈ 𝑊)

Assertion

Ref	Expression
frecabex	⊢ (𝜑 → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚))) ∨ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴))} ∈ 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

Step	Hyp	Ref	Expression
1		omex 4715	. . . 4 ⊢ ω ∈ V
2		simpr 110	. . . . . . 7 ⊢ ((dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚))) → 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))
3	2	abssi 3313	. . . . . 6 ⊢ {𝑥 ∣ (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))} ⊆ (𝐹‘(𝑆‘𝑚))
4		frecabex.sex	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
5		vex 2816	. . . . . . . 8 ⊢ 𝑚 ∈ V
6		fvexg 5689	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑚 ∈ V) → (𝑆‘𝑚) ∈ V)
7	4, 5, 6	sylancl 413	. . . . . . 7 ⊢ (𝜑 → (𝑆‘𝑚) ∈ V)
8		frecabex.fvex	. . . . . . 7 ⊢ (𝜑 → ∀𝑦(𝐹‘𝑦) ∈ V)
9		fveq2 5670	. . . . . . . . 9 ⊢ (𝑦 = (𝑆‘𝑚) → (𝐹‘𝑦) = (𝐹‘(𝑆‘𝑚)))
10	9	eleq1d 2301	. . . . . . . 8 ⊢ (𝑦 = (𝑆‘𝑚) → ((𝐹‘𝑦) ∈ V ↔ (𝐹‘(𝑆‘𝑚)) ∈ V))
11	10	spcgv 2904	. . . . . . 7 ⊢ ((𝑆‘𝑚) ∈ V → (∀𝑦(𝐹‘𝑦) ∈ V → (𝐹‘(𝑆‘𝑚)) ∈ V))
12	7, 8, 11	sylc 62	. . . . . 6 ⊢ (𝜑 → (𝐹‘(𝑆‘𝑚)) ∈ V)
13		ssexg 4249	. . . . . 6 ⊢ (({𝑥 ∣ (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))} ⊆ (𝐹‘(𝑆‘𝑚)) ∧ (𝐹‘(𝑆‘𝑚)) ∈ V) → {𝑥 ∣ (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))} ∈ V)
14	3, 12, 13	sylancr 414	. . . . 5 ⊢ (𝜑 → {𝑥 ∣ (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))} ∈ V)
15	14	ralrimivw 2616	. . . 4 ⊢ (𝜑 → ∀𝑚 ∈ ω {𝑥 ∣ (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))} ∈ V)
16		abrexex2g 6313	. . . 4 ⊢ ((ω ∈ V ∧ ∀𝑚 ∈ ω {𝑥 ∣ (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))} ∈ V) → {𝑥 ∣ ∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))} ∈ V)
17	1, 15, 16	sylancr 414	. . 3 ⊢ (𝜑 → {𝑥 ∣ ∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))} ∈ V)
18		simpr 110	. . . . 5 ⊢ ((dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
19	18	abssi 3313	. . . 4 ⊢ {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴)} ⊆ 𝐴
20		frecabex.aex	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑊)
21		ssexg 4249	. . . 4 ⊢ (({𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴)} ⊆ 𝐴 ∧ 𝐴 ∈ 𝑊) → {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴)} ∈ V)
22	19, 20, 21	sylancr 414	. . 3 ⊢ (𝜑 → {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴)} ∈ V)
23	17, 22	jca 306	. 2 ⊢ (𝜑 → ({𝑥 ∣ ∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))} ∈ V ∧ {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴)} ∈ V))
24		unexb 4563	. . 3 ⊢ (({𝑥 ∣ ∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))} ∈ V ∧ {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴)} ∈ V) ↔ ({𝑥 ∣ ∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))} ∪ {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴)}) ∈ V)
25		unab 3488	. . . 4 ⊢ ({𝑥 ∣ ∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))} ∪ {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴)}) = {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚))) ∨ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴))}
26	25	eleq1i 2298	. . 3 ⊢ (({𝑥 ∣ ∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))} ∪ {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴)}) ∈ V ↔ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚))) ∨ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴))} ∈ V)
27	24, 26	bitri 184	. 2 ⊢ (({𝑥 ∣ ∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))} ∈ V ∧ {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴)} ∈ V) ↔ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚))) ∨ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴))} ∈ V)
28	23, 27	sylib 122	1 ⊢ (𝜑 → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚))) ∨ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴))} ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 716  ∀wal 1396   = wceq 1398   ∈ wcel 2203  {cab 2218  ∀wral 2520  ∃wrex 2521  Vcvv 2813   ∪ cun 3209   ⊆ wss 3211  ∅c0 3508  suc csuc 4486  ωcom 4712  dom cdm 4749  ‘cfv 5352

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360

This theorem is referenced by:  frectfr  6631