Theorem frecabex 6249

Description: The class abstraction from df-frec 6242 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 4467	. . . 4 ⊢ ω ∈ V
2		simpr 109	. . . . . . 7 ⊢ ((dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚))) → 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))
3	2	abssi 3138	. . . . . 6 ⊢ {𝑥 ∣ (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))} ⊆ (𝐹‘(𝑆‘𝑚))
4		frecabex.sex	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
5		vex 2660	. . . . . . . 8 ⊢ 𝑚 ∈ V
6		fvexg 5394	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑚 ∈ V) → (𝑆‘𝑚) ∈ V)
7	4, 5, 6	sylancl 407	. . . . . . 7 ⊢ (𝜑 → (𝑆‘𝑚) ∈ V)
8		frecabex.fvex	. . . . . . 7 ⊢ (𝜑 → ∀𝑦(𝐹‘𝑦) ∈ V)
9		fveq2 5375	. . . . . . . . 9 ⊢ (𝑦 = (𝑆‘𝑚) → (𝐹‘𝑦) = (𝐹‘(𝑆‘𝑚)))
10	9	eleq1d 2183	. . . . . . . 8 ⊢ (𝑦 = (𝑆‘𝑚) → ((𝐹‘𝑦) ∈ V ↔ (𝐹‘(𝑆‘𝑚)) ∈ V))
11	10	spcgv 2744	. . . . . . 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 4027	. . . . . 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 408	. . . . 5 ⊢ (𝜑 → {𝑥 ∣ (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))} ∈ V)
15	14	ralrimivw 2480	. . . 4 ⊢ (𝜑 → ∀𝑚 ∈ ω {𝑥 ∣ (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))} ∈ V)
16		abrexex2g 5972	. . . 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 408	. . 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 3138	. . . 4 ⊢ {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴)} ⊆ 𝐴
20		frecabex.aex	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑊)
21		ssexg 4027	. . . 4 ⊢ (({𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴)} ⊆ 𝐴 ∧ 𝐴 ∈ 𝑊) → {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴)} ∈ V)
22	19, 20, 21	sylancr 408	. . 3 ⊢ (𝜑 → {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴)} ∈ V)
23	17, 22	jca 302	. 2 ⊢ (𝜑 → ({𝑥 ∣ ∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚)))} ∈ V ∧ {𝑥 ∣ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴)} ∈ V))
24		unexb 4323	. . 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 3309	. . . 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 2180	. . 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 680  ∀wal 1312   = wceq 1314   ∈ wcel 1463  {cab 2101  ∀wral 2390  ∃wrex 2391  Vcvv 2657   ∪ cun 3035   ⊆ wss 3037  ∅c0 3329  suc csuc 4247  ωcom 4464  dom cdm 4499  ‘cfv 5081

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-iinf 4462

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089

This theorem is referenced by:  frectfr  6251