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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 (φ𝑆 𝑉)
frecabex.2 (φ𝐹 Fn V)
frecabex.3 (φA 𝑊)
Assertion
Ref Expression
frecabex (φ → {x ∣ (𝑚 𝜔 (dom 𝑆 = suc 𝑚 x (𝐹‘(𝑆𝑚))) (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
StepHypRef Expression
1 omex 4243 . . . 4 𝜔 V
2 ax-ia2 100 . . . . . . 7 ((dom 𝑆 = suc 𝑚 x (𝐹‘(𝑆𝑚))) → x (𝐹‘(𝑆𝑚)))
32abssi 2992 . . . . . 6 {x ∣ (dom 𝑆 = suc 𝑚 x (𝐹‘(𝑆𝑚)))} ⊆ (𝐹‘(𝑆𝑚))
4 frecabex.2 . . . . . . 7 (φ𝐹 Fn V)
5 frecabex.1 . . . . . . . 8 (φ𝑆 𝑉)
6 vex 2538 . . . . . . . 8 𝑚 V
7 fvexg 5119 . . . . . . . 8 ((𝑆 𝑉 𝑚 V) → (𝑆𝑚) V)
85, 6, 7sylancl 394 . . . . . . 7 (φ → (𝑆𝑚) V)
9 funfvex 5117 . . . . . . . 8 ((Fun 𝐹 (𝑆𝑚) dom 𝐹) → (𝐹‘(𝑆𝑚)) V)
109funfni 4925 . . . . . . 7 ((𝐹 Fn V (𝑆𝑚) V) → (𝐹‘(𝑆𝑚)) V)
114, 8, 10syl2anc 393 . . . . . 6 (φ → (𝐹‘(𝑆𝑚)) V)
12 ssexg 3870 . . . . . 6 (({x ∣ (dom 𝑆 = suc 𝑚 x (𝐹‘(𝑆𝑚)))} ⊆ (𝐹‘(𝑆𝑚)) (𝐹‘(𝑆𝑚)) V) → {x ∣ (dom 𝑆 = suc 𝑚 x (𝐹‘(𝑆𝑚)))} V)
133, 11, 12sylancr 395 . . . . 5 (φ → {x ∣ (dom 𝑆 = suc 𝑚 x (𝐹‘(𝑆𝑚)))} V)
1413ralrimivw 2371 . . . 4 (φ𝑚 𝜔 {x ∣ (dom 𝑆 = suc 𝑚 x (𝐹‘(𝑆𝑚)))} V)
15 abrexex2g 5670 . . . 4 ((𝜔 V 𝑚 𝜔 {x ∣ (dom 𝑆 = suc 𝑚 x (𝐹‘(𝑆𝑚)))} V) → {x𝑚 𝜔 (dom 𝑆 = suc 𝑚 x (𝐹‘(𝑆𝑚)))} V)
161, 14, 15sylancr 395 . . 3 (φ → {x𝑚 𝜔 (dom 𝑆 = suc 𝑚 x (𝐹‘(𝑆𝑚)))} V)
17 ax-ia2 100 . . . . 5 ((dom 𝑆 = ∅ x A) → x A)
1817abssi 2992 . . . 4 {x ∣ (dom 𝑆 = ∅ x A)} ⊆ A
19 frecabex.3 . . . 4 (φA 𝑊)
20 ssexg 3870 . . . 4 (({x ∣ (dom 𝑆 = ∅ x A)} ⊆ A A 𝑊) → {x ∣ (dom 𝑆 = ∅ x A)} V)
2118, 19, 20sylancr 395 . . 3 (φ → {x ∣ (dom 𝑆 = ∅ x A)} V)
2216, 21jca 290 . 2 (φ → ({x𝑚 𝜔 (dom 𝑆 = suc 𝑚 x (𝐹‘(𝑆𝑚)))} V {x ∣ (dom 𝑆 = ∅ x A)} V))
23 unexb 4127 . . 3 (({x𝑚 𝜔 (dom 𝑆 = suc 𝑚 x (𝐹‘(𝑆𝑚)))} V {x ∣ (dom 𝑆 = ∅ x A)} V) ↔ ({x𝑚 𝜔 (dom 𝑆 = suc 𝑚 x (𝐹‘(𝑆𝑚)))} ∪ {x ∣ (dom 𝑆 = ∅ x A)}) V)
24 unab 3181 . . . 4 ({x𝑚 𝜔 (dom 𝑆 = suc 𝑚 x (𝐹‘(𝑆𝑚)))} ∪ {x ∣ (dom 𝑆 = ∅ x A)}) = {x ∣ (𝑚 𝜔 (dom 𝑆 = suc 𝑚 x (𝐹‘(𝑆𝑚))) (dom 𝑆 = ∅ x A))}
2524eleq1i 2085 . . 3 (({x𝑚 𝜔 (dom 𝑆 = suc 𝑚 x (𝐹‘(𝑆𝑚)))} ∪ {x ∣ (dom 𝑆 = ∅ x A)}) V ↔ {x ∣ (𝑚 𝜔 (dom 𝑆 = suc 𝑚 x (𝐹‘(𝑆𝑚))) (dom 𝑆 = ∅ x A))} V)
2623, 25bitri 173 . 2 (({x𝑚 𝜔 (dom 𝑆 = suc 𝑚 x (𝐹‘(𝑆𝑚)))} V {x ∣ (dom 𝑆 = ∅ x A)} V) ↔ {x ∣ (𝑚 𝜔 (dom 𝑆 = suc 𝑚 x (𝐹‘(𝑆𝑚))) (dom 𝑆 = ∅ x A))} V)
2722, 26sylib 127 1 (φ → {x ∣ (𝑚 𝜔 (dom 𝑆 = suc 𝑚 x (𝐹‘(𝑆𝑚))) (dom 𝑆 = ∅ x A))} V)
Colors of variables: wff set class
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
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