Theorem frecabex 6563

Description: The class abstraction from df-frec 6556 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 4691	. . . 4 ⊢ ω ∈ V
2		simpr 110	. . . . . . 7 ⊢ ((dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚))) → 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))
3	2	abssi 3302	. . . . . 6 ⊢ {𝑥 ∣ (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))} ⊆ (𝐹‘(𝑆‘𝑚))
4		frecabex.sex	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
5		vex 2805	. . . . . . . 8 ⊢ 𝑚 ∈ V
6		fvexg 5658	. . . . . . . 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 5639	. . . . . . . . 9 ⊢ (𝑦 = (𝑆‘𝑚) → (𝐹‘𝑦) = (𝐹‘(𝑆‘𝑚)))
10	9	eleq1d 2300	. . . . . . . 8 ⊢ (𝑦 = (𝑆‘𝑚) → ((𝐹‘𝑦) ∈ V ↔ (𝐹‘(𝑆‘𝑚)) ∈ V))
11	10	spcgv 2893	. . . . . . 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 4228	. . . . . 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 2606	. . . 4 ⊢ (𝜑 → ∀𝑚 ∈ ω {𝑥 ∣ (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))} ∈ V)
16		abrexex2g 6281	. . . 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 3302	. . . 4 ⊢ {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴)} ⊆ 𝐴
20		frecabex.aex	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑊)
21		ssexg 4228	. . . 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 4539	. . 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 3474	. . . 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 2297	. . 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 715  ∀wal 1395   = wceq 1397   ∈ wcel 2202  {cab 2217  ∀wral 2510  ∃wrex 2511  Vcvv 2802   ∪ cun 3198   ⊆ wss 3200  ∅c0 3494  suc csuc 4462  ωcom 4688  dom cdm 4725  ‘cfv 5326

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334

This theorem is referenced by:  frectfr  6565