Theorem frecabex 5899

Description: The class abstraction from df-frec 5898 exists. This is a lemma for several other finite recursion proofs. (Contributed by Jim Kingdon, 16-Aug-2019.)

Hypotheses

Ref	Expression
frecabex.1	⊢ (φ → 𝑆 ∈ 𝑉)
frecabex.2	⊢ (φ → 𝐹 Fn V)
frecabex.3	⊢ (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   φ,𝑚   x,𝑚

Allowed substitution hints:   φ(x)   A(𝑚)   𝑆(𝑚)   𝐹(𝑚)   𝑉(x,𝑚)   𝑊(x,𝑚)

Proof of Theorem frecabex

Step	Hyp	Ref	Expression
1		omex 4243	. . . 4 ⊢ 𝜔 ∈ V
2		ax-ia2 100	. . . . . . 7 ⊢ ((dom 𝑆 = suc 𝑚 ∧ x ∈ (𝐹‘(𝑆‘𝑚))) → x ∈ (𝐹‘(𝑆‘𝑚)))
3	2	abssi 2992	. . . . . 6 ⊢ {x ∣ (dom 𝑆 = suc 𝑚 ∧ x ∈ (𝐹‘(𝑆‘𝑚)))} ⊆ (𝐹‘(𝑆‘𝑚))
4		frecabex.2	. . . . . . 7 ⊢ (φ → 𝐹 Fn V)
5		frecabex.1	. . . . . . . 8 ⊢ (φ → 𝑆 ∈ 𝑉)
6		vex 2538	. . . . . . . 8 ⊢ 𝑚 ∈ V
7		fvexg 5119	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑚 ∈ V) → (𝑆‘𝑚) ∈ V)
8	5, 6, 7	sylancl 394	. . . . . . 7 ⊢ (φ → (𝑆‘𝑚) ∈ V)
9		funfvex 5117	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ (𝑆‘𝑚) ∈ dom 𝐹) → (𝐹‘(𝑆‘𝑚)) ∈ V)
10	9	funfni 4925	. . . . . . 7 ⊢ ((𝐹 Fn V ∧ (𝑆‘𝑚) ∈ V) → (𝐹‘(𝑆‘𝑚)) ∈ V)
11	4, 8, 10	syl2anc 393	. . . . . 6 ⊢ (φ → (𝐹‘(𝑆‘𝑚)) ∈ V)
12		ssexg 3870	. . . . . 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)
13	3, 11, 12	sylancr 395	. . . . 5 ⊢ (φ → {x ∣ (dom 𝑆 = suc 𝑚 ∧ x ∈ (𝐹‘(𝑆‘𝑚)))} ∈ V)
14	13	ralrimivw 2371	. . . 4 ⊢ (φ → ∀𝑚 ∈ 𝜔 {x ∣ (dom 𝑆 = suc 𝑚 ∧ x ∈ (𝐹‘(𝑆‘𝑚)))} ∈ V)
15		abrexex2g 5670	. . . 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)
16	1, 14, 15	sylancr 395	. . 3 ⊢ (φ → {x ∣ ∃𝑚 ∈ 𝜔 (dom 𝑆 = suc 𝑚 ∧ x ∈ (𝐹‘(𝑆‘𝑚)))} ∈ V)
17		ax-ia2 100	. . . . 5 ⊢ ((dom 𝑆 = ∅ ∧ x ∈ A) → x ∈ A)
18	17	abssi 2992	. . . 4 ⊢ {x ∣ (dom 𝑆 = ∅ ∧ x ∈ A)} ⊆ A
19		frecabex.3	. . . 4 ⊢ (φ → A ∈ 𝑊)
20		ssexg 3870	. . . 4 ⊢ (({x ∣ (dom 𝑆 = ∅ ∧ x ∈ A)} ⊆ A ∧ A ∈ 𝑊) → {x ∣ (dom 𝑆 = ∅ ∧ x ∈ A)} ∈ V)
21	18, 19, 20	sylancr 395	. . 3 ⊢ (φ → {x ∣ (dom 𝑆 = ∅ ∧ x ∈ A)} ∈ V)
22	16, 21	jca 290	. 2 ⊢ (φ → ({x ∣ ∃𝑚 ∈ 𝜔 (dom 𝑆 = suc 𝑚 ∧ x ∈ (𝐹‘(𝑆‘𝑚)))} ∈ V ∧ {x ∣ (dom 𝑆 = ∅ ∧ x ∈ A)} ∈ V))
23		unexb 4127	. . 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)
24		unab 3181	. . . 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))}
25	24	eleq1i 2085	. . 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)
26	23, 25	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)
27	22, 26	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 616   = wceq 1228   ∈ wcel 1374  {cab 2008  ∀wral 2284  ∃wrex 2285  Vcvv 2535   ∪ cun 2892   ⊆ wss 2894  ∅c0 3201  suc csuc 4051  𝜔com 4240  dom cdm 4272   Fn wfn 4824  ‘cfv 4829

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-iinf 4238

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-int 3590  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-iom 4241  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837

This theorem is referenced by:  frectfr  5900  frecsuclem3  5906