Theorem frecabex 6366

Description: The class abstraction from df-frec 6359 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 4570	. . . 4 ⊢ ω ∈ V
2		simpr 109	. . . . . . 7 ⊢ ((dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚))) → 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))
3	2	abssi 3217	. . . . . 6 ⊢ {𝑥 ∣ (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))} ⊆ (𝐹‘(𝑆‘𝑚))
4		frecabex.sex	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
5		vex 2729	. . . . . . . 8 ⊢ 𝑚 ∈ V
6		fvexg 5505	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑚 ∈ V) → (𝑆‘𝑚) ∈ V)
7	4, 5, 6	sylancl 410	. . . . . . 7 ⊢ (𝜑 → (𝑆‘𝑚) ∈ V)
8		frecabex.fvex	. . . . . . 7 ⊢ (𝜑 → ∀𝑦(𝐹‘𝑦) ∈ V)
9		fveq2 5486	. . . . . . . . 9 ⊢ (𝑦 = (𝑆‘𝑚) → (𝐹‘𝑦) = (𝐹‘(𝑆‘𝑚)))
10	9	eleq1d 2235	. . . . . . . 8 ⊢ (𝑦 = (𝑆‘𝑚) → ((𝐹‘𝑦) ∈ V ↔ (𝐹‘(𝑆‘𝑚)) ∈ V))
11	10	spcgv 2813	. . . . . . 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 4121	. . . . . 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 411	. . . . 5 ⊢ (𝜑 → {𝑥 ∣ (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))} ∈ V)
15	14	ralrimivw 2540	. . . 4 ⊢ (𝜑 → ∀𝑚 ∈ ω {𝑥 ∣ (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))} ∈ V)
16		abrexex2g 6088	. . . 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 411	. . 3 ⊢ (𝜑 → {𝑥 ∣ ∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))} ∈ V)
18		simpr 109	. . . . 5 ⊢ ((dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
19	18	abssi 3217	. . . 4 ⊢ {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴)} ⊆ 𝐴
20		frecabex.aex	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑊)
21		ssexg 4121	. . . 4 ⊢ (({𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴)} ⊆ 𝐴 ∧ 𝐴 ∈ 𝑊) → {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴)} ∈ V)
22	19, 20, 21	sylancr 411	. . 3 ⊢ (𝜑 → {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴)} ∈ V)
23	17, 22	jca 304	. 2 ⊢ (𝜑 → ({𝑥 ∣ ∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))} ∈ V ∧ {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴)} ∈ V))
24		unexb 4420	. . 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 3389	. . . 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 2232	. . 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 183	. 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 121	1 ⊢ (𝜑 → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚))) ∨ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴))} ∈ V)

Syntax hints:   → wi 4   ∧ wa 103   ∨ wo 698  ∀wal 1341   = wceq 1343   ∈ wcel 2136  {cab 2151  ∀wral 2444  ∃wrex 2445  Vcvv 2726   ∪ cun 3114   ⊆ wss 3116  ∅c0 3409  suc csuc 4343  ωcom 4567  dom cdm 4604  ‘cfv 5188

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196

This theorem is referenced by:  frectfr  6368