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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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