Theorem frecabex 6642

Description: The class abstraction from df-frec 6635 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 4720	. . . 4 ⊢ ω ∈ V
2		simpr 110	. . . . . . 7 ⊢ ((dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚))) → 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))
3	2	abssi 3317	. . . . . 6 ⊢ {𝑥 ∣ (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))} ⊆ (𝐹‘(𝑆‘𝑚))
4		frecabex.sex	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
5		vex 2818	. . . . . . . 8 ⊢ 𝑚 ∈ V
6		fvexg 5694	. . . . . . . 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 5675	. . . . . . . . 9 ⊢ (𝑦 = (𝑆‘𝑚) → (𝐹‘𝑦) = (𝐹‘(𝑆‘𝑚)))
10	9	eleq1d 2303	. . . . . . . 8 ⊢ (𝑦 = (𝑆‘𝑚) → ((𝐹‘𝑦) ∈ V ↔ (𝐹‘(𝑆‘𝑚)) ∈ V))
11	10	spcgv 2906	. . . . . . 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 4254	. . . . . 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 2618	. . . 4 ⊢ (𝜑 → ∀𝑚 ∈ ω {𝑥 ∣ (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))} ∈ V)
16		abrexex2g 6322	. . . 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 3317	. . . 4 ⊢ {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴)} ⊆ 𝐴
20		frecabex.aex	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑊)
21		ssexg 4254	. . . 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 4568	. . 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 3492	. . . 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 2300	. . 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 2205  {cab 2220  ∀wral 2522  ∃wrex 2523  Vcvv 2815   ∪ cun 3212   ⊆ wss 3214  ∅c0 3512  suc csuc 4491  ωcom 4717  dom cdm 4754  ‘cfv 5357

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365

This theorem is referenced by:  frectfr  6644