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Theorem frec0g 5902
 Description: The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 11-Aug-2019.)
Assertion
Ref Expression
frec0g ((𝐹 Fn V A 𝑉) → (frec(𝐹, A)‘∅) = A)

Proof of Theorem frec0g
Dummy variables g 𝑚 x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dm0 4476 . . . . . . . . 9 dom ∅ = ∅
21biantrur 287 . . . . . . . 8 (x A ↔ (dom ∅ = ∅ x A))
3 vex 2538 . . . . . . . . . . . . . 14 𝑚 V
4 nsuceq0g 4104 . . . . . . . . . . . . . 14 (𝑚 V → suc 𝑚 ≠ ∅)
53, 4ax-mp 7 . . . . . . . . . . . . 13 suc 𝑚 ≠ ∅
6 eqcom 2024 . . . . . . . . . . . . . 14 (suc 𝑚 = dom ∅ ↔ dom ∅ = suc 𝑚)
71eqeq2i 2032 . . . . . . . . . . . . . 14 (suc 𝑚 = dom ∅ ↔ suc 𝑚 = ∅)
86, 7bitr3i 175 . . . . . . . . . . . . 13 (dom ∅ = suc 𝑚 ↔ suc 𝑚 = ∅)
95, 8nemtbir 2272 . . . . . . . . . . . 12 ¬ dom ∅ = suc 𝑚
109intnanr 827 . . . . . . . . . . 11 ¬ (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚)))
1110a1i 9 . . . . . . . . . 10 (𝑚 𝜔 → ¬ (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))))
1211nrex 2389 . . . . . . . . 9 ¬ 𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚)))
1312biorfi 652 . . . . . . . 8 ((dom ∅ = ∅ x A) ↔ ((dom ∅ = ∅ x A) 𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚)))))
14 orcom 634 . . . . . . . 8 (((dom ∅ = ∅ x A) 𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚)))) ↔ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A)))
152, 13, 143bitri 195 . . . . . . 7 (x A ↔ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A)))
1615abbii 2135 . . . . . 6 {xx A} = {x ∣ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A))}
17 abid2 2140 . . . . . 6 {xx A} = A
1816, 17eqtr3i 2044 . . . . 5 {x ∣ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A))} = A
1918eleq1i 2085 . . . 4 ({x ∣ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A))} 𝑉A 𝑉)
2019biimpri 124 . . 3 (A 𝑉 → {x ∣ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A))} 𝑉)
2120adantl 262 . 2 ((𝐹 Fn V A 𝑉) → {x ∣ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A))} 𝑉)
22 0ex 3858 . . . . 5 V
23 dmeq 4462 . . . . . . . . . . 11 (g = ∅ → dom g = dom ∅)
2423eqeq1d 2030 . . . . . . . . . 10 (g = ∅ → (dom g = suc 𝑚 ↔ dom ∅ = suc 𝑚))
25 fveq1 5102 . . . . . . . . . . . 12 (g = ∅ → (g𝑚) = (∅‘𝑚))
2625fveq2d 5107 . . . . . . . . . . 11 (g = ∅ → (𝐹‘(g𝑚)) = (𝐹‘(∅‘𝑚)))
2726eleq2d 2089 . . . . . . . . . 10 (g = ∅ → (x (𝐹‘(g𝑚)) ↔ x (𝐹‘(∅‘𝑚))))
2824, 27anbi12d 445 . . . . . . . . 9 (g = ∅ → ((dom g = suc 𝑚 x (𝐹‘(g𝑚))) ↔ (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚)))))
2928rexbidv 2305 . . . . . . . 8 (g = ∅ → (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) ↔ 𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚)))))
3023eqeq1d 2030 . . . . . . . . 9 (g = ∅ → (dom g = ∅ ↔ dom ∅ = ∅))
3130anbi1d 441 . . . . . . . 8 (g = ∅ → ((dom g = ∅ x A) ↔ (dom ∅ = ∅ x A)))
3229, 31orbi12d 694 . . . . . . 7 (g = ∅ → ((𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A)) ↔ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A))))
3332abbidv 2137 . . . . . 6 (g = ∅ → {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))} = {x ∣ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A))})
34 eqid 2022 . . . . . 6 (g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))}) = (g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})
3533, 34fvmptg 5173 . . . . 5 ((∅ V {x ∣ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A))} 𝑉) → ((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})‘∅) = {x ∣ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A))})
3622, 35mpan 402 . . . 4 ({x ∣ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A))} 𝑉 → ((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})‘∅) = {x ∣ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A))})
37 df-frec 5898 . . . . . . . . 9 frec(𝐹, A) = (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) ↾ 𝜔)
3837fveq1i 5104 . . . . . . . 8 (frec(𝐹, A)‘∅) = ((recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) ↾ 𝜔)‘∅)
39 peano1 4244 . . . . . . . . 9 𝜔
40 fvres 5123 . . . . . . . . 9 (∅ 𝜔 → ((recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) ↾ 𝜔)‘∅) = (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))}))‘∅))
4139, 40ax-mp 7 . . . . . . . 8 ((recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) ↾ 𝜔)‘∅) = (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))}))‘∅)
4238, 41eqtri 2042 . . . . . . 7 (frec(𝐹, A)‘∅) = (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))}))‘∅)
43 0elon 4078 . . . . . . . 8 On
44 eqid 2022 . . . . . . . . 9 recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) = recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))}))
4534frectfr 5900 . . . . . . . . 9 ((𝐹 Fn V A 𝑉) → y(Fun (g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))}) ((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})‘y) V))
4644, 45tfri2d 5872 . . . . . . . 8 (((𝐹 Fn V A 𝑉) On) → (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))}))‘∅) = ((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})‘(recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) ↾ ∅)))
4743, 46mpan2 403 . . . . . . 7 ((𝐹 Fn V A 𝑉) → (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))}))‘∅) = ((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})‘(recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) ↾ ∅)))
4842, 47syl5eq 2066 . . . . . 6 ((𝐹 Fn V A 𝑉) → (frec(𝐹, A)‘∅) = ((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})‘(recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) ↾ ∅)))
49 res0 4543 . . . . . . 7 (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) ↾ ∅) = ∅
5049fveq2i 5106 . . . . . 6 ((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})‘(recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) ↾ ∅)) = ((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})‘∅)
5148, 50syl6eq 2070 . . . . 5 ((𝐹 Fn V A 𝑉) → (frec(𝐹, A)‘∅) = ((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})‘∅))
5251eqeq1d 2030 . . . 4 ((𝐹 Fn V A 𝑉) → ((frec(𝐹, A)‘∅) = {x ∣ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A))} ↔ ((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})‘∅) = {x ∣ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A))}))
5336, 52syl5ibr 145 . . 3 ((𝐹 Fn V A 𝑉) → ({x ∣ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A))} 𝑉 → (frec(𝐹, A)‘∅) = {x ∣ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A))}))
5418eqeq2i 2032 . . 3 ((frec(𝐹, A)‘∅) = {x ∣ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A))} ↔ (frec(𝐹, A)‘∅) = A)
5553, 54syl6ib 150 . 2 ((𝐹 Fn V A 𝑉) → ({x ∣ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A))} 𝑉 → (frec(𝐹, A)‘∅) = A))
5621, 55mpd 13 1 ((𝐹 Fn V A 𝑉) → (frec(𝐹, A)‘∅) = A)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∨ wo 616   = wceq 1228   ∈ wcel 1374  {cab 2008   ≠ wne 2186  ∃wrex 2285  Vcvv 2535  ∅c0 3201   ↦ cmpt 3792  Oncon0 4049  suc csuc 4051  𝜔com 4240  dom cdm 4272   ↾ cres 4274   Fn wfn 4824  ‘cfv 4829  recscrecs 5841  freccfrec 5897 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-in1 532  ax-in2 533  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-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204  ax-iinf 4238 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  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-ne 2188  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  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-tr 3829  df-id 4004  df-iord 4052  df-on 4054  df-suc 4057  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  df-recs 5842  df-frec 5898 This theorem is referenced by:  frecrdg  5908
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