Theorem frecabex 5923

Description: The class abstraction from df-frec 5918 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 4259	. . . 4 ⊢ 𝜔 ∈ V
2		simpr 103	. . . . . . 7 ⊢ ((dom 𝑆 = suc 𝑚 ∧ x ∈ (𝐹‘(𝑆‘𝑚))) → x ∈ (𝐹‘(𝑆‘𝑚)))
3	2	abssi 3009	. . . . . 6 ⊢ {x ∣ (dom 𝑆 = suc 𝑚 ∧ x ∈ (𝐹‘(𝑆‘𝑚)))} ⊆ (𝐹‘(𝑆‘𝑚))
4		frecabex.sex	. . . . . . . 8 ⊢ (φ → 𝑆 ∈ 𝑉)
5		vex 2554	. . . . . . . 8 ⊢ 𝑚 ∈ V
6		fvexg 5137	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑚 ∈ V) → (𝑆‘𝑚) ∈ V)
7	4, 5, 6	sylancl 392	. . . . . . 7 ⊢ (φ → (𝑆‘𝑚) ∈ V)
8		frecabex.fvex	. . . . . . 7 ⊢ (φ → ∀y(𝐹‘y) ∈ V)
9		fveq2 5121	. . . . . . . . 9 ⊢ (y = (𝑆‘𝑚) → (𝐹‘y) = (𝐹‘(𝑆‘𝑚)))
10	9	eleq1d 2103	. . . . . . . 8 ⊢ (y = (𝑆‘𝑚) → ((𝐹‘y) ∈ V ↔ (𝐹‘(𝑆‘𝑚)) ∈ V))
11	10	spcgv 2634	. . . . . . 7 ⊢ ((𝑆‘𝑚) ∈ V → (∀y(𝐹‘y) ∈ V → (𝐹‘(𝑆‘𝑚)) ∈ V))
12	7, 8, 11	sylc 56	. . . . . 6 ⊢ (φ → (𝐹‘(𝑆‘𝑚)) ∈ V)
13		ssexg 3887	. . . . . 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 393	. . . . 5 ⊢ (φ → {x ∣ (dom 𝑆 = suc 𝑚 ∧ x ∈ (𝐹‘(𝑆‘𝑚)))} ∈ V)
15	14	ralrimivw 2387	. . . 4 ⊢ (φ → ∀𝑚 ∈ 𝜔 {x ∣ (dom 𝑆 = suc 𝑚 ∧ x ∈ (𝐹‘(𝑆‘𝑚)))} ∈ V)
16		abrexex2g 5689	. . . 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 393	. . 3 ⊢ (φ → {x ∣ ∃𝑚 ∈ 𝜔 (dom 𝑆 = suc 𝑚 ∧ x ∈ (𝐹‘(𝑆‘𝑚)))} ∈ V)
18		simpr 103	. . . . 5 ⊢ ((dom 𝑆 = ∅ ∧ x ∈ A) → x ∈ A)
19	18	abssi 3009	. . . 4 ⊢ {x ∣ (dom 𝑆 = ∅ ∧ x ∈ A)} ⊆ A
20		frecabex.aex	. . . 4 ⊢ (φ → A ∈ 𝑊)
21		ssexg 3887	. . . 4 ⊢ (({x ∣ (dom 𝑆 = ∅ ∧ x ∈ A)} ⊆ A ∧ A ∈ 𝑊) → {x ∣ (dom 𝑆 = ∅ ∧ x ∈ A)} ∈ V)
22	19, 20, 21	sylancr 393	. . 3 ⊢ (φ → {x ∣ (dom 𝑆 = ∅ ∧ x ∈ A)} ∈ V)
23	17, 22	jca 290	. 2 ⊢ (φ → ({x ∣ ∃𝑚 ∈ 𝜔 (dom 𝑆 = suc 𝑚 ∧ x ∈ (𝐹‘(𝑆‘𝑚)))} ∈ V ∧ {x ∣ (dom 𝑆 = ∅ ∧ x ∈ A)} ∈ V))
24		unexb 4143	. . 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 3198	. . . 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 2100	. . 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 173	. 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 127	1 ⊢ (φ → {x ∣ (∃𝑚 ∈ 𝜔 (dom 𝑆 = suc 𝑚 ∧ x ∈ (𝐹‘(𝑆‘𝑚))) ∨ (dom 𝑆 = ∅ ∧ x ∈ A))} ∈ V)

Syntax hints:   → wi 4   ∧ wa 97   ∨ wo 628  ∀wal 1240   = wceq 1242   ∈ wcel 1390  {cab 2023  ∀wral 2300  ∃wrex 2301  Vcvv 2551   ∪ cun 2909   ⊆ wss 2911  ∅c0 3218  suc csuc 4068  𝜔com 4256  dom cdm 4288  ‘cfv 4845

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853

This theorem is referenced by:  frectfr  5924  frecsuclem3  5929