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Theorem rdgi0g 5882
Description: The initial value of the recursive definition generator. (Contributed by Jim Kingdon, 10-Jul-2019.)
Assertion
Ref Expression
rdgi0g ((𝐹 Fn V A 𝑉) → (rec(𝐹, A)‘∅) = A)

Proof of Theorem rdgi0g
Dummy variables g x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0elon 4074 . . . . 5 On
2 df-irdg 5874 . . . . . 6 rec(𝐹, A) = recs((g V ↦ (A x dom g(𝐹‘(gx)))))
3 rdgruledefgg 5878 . . . . . . 7 ((𝐹 Fn V A 𝑉) → (Fun (g V ↦ (A x dom g(𝐹‘(gx)))) ((g V ↦ (A x dom g(𝐹‘(gx))))‘y) V))
43alrimiv 1732 . . . . . 6 ((𝐹 Fn V A 𝑉) → y(Fun (g V ↦ (A x dom g(𝐹‘(gx)))) ((g V ↦ (A x dom g(𝐹‘(gx))))‘y) V))
52, 4tfri2d 5868 . . . . 5 (((𝐹 Fn V A 𝑉) On) → (rec(𝐹, A)‘∅) = ((g V ↦ (A x dom g(𝐹‘(gx))))‘(rec(𝐹, A) ↾ ∅)))
61, 5mpan2 403 . . . 4 ((𝐹 Fn V A 𝑉) → (rec(𝐹, A)‘∅) = ((g V ↦ (A x dom g(𝐹‘(gx))))‘(rec(𝐹, A) ↾ ∅)))
7 res0 4539 . . . . 5 (rec(𝐹, A) ↾ ∅) = ∅
87fveq2i 5102 . . . 4 ((g V ↦ (A x dom g(𝐹‘(gx))))‘(rec(𝐹, A) ↾ ∅)) = ((g V ↦ (A x dom g(𝐹‘(gx))))‘∅)
96, 8syl6eq 2066 . . 3 ((𝐹 Fn V A 𝑉) → (rec(𝐹, A)‘∅) = ((g V ↦ (A x dom g(𝐹‘(gx))))‘∅))
10 0ex 3854 . . . 4 V
1110dmex 4521 . . . . . 6 dom ∅ V
12 0fv 5129 . . . . . . . . 9 (∅‘x) = ∅
1312fveq2i 5102 . . . . . . . 8 (𝐹‘(∅‘x)) = (𝐹‘∅)
14 funfvex 5113 . . . . . . . . . 10 ((Fun 𝐹 dom 𝐹) → (𝐹‘∅) V)
1514funfni 4921 . . . . . . . . 9 ((𝐹 Fn V V) → (𝐹‘∅) V)
1610, 15mpan2 403 . . . . . . . 8 (𝐹 Fn V → (𝐹‘∅) V)
1713, 16syl5eqel 2102 . . . . . . 7 (𝐹 Fn V → (𝐹‘(∅‘x)) V)
1817ralrimivw 2367 . . . . . 6 (𝐹 Fn V → x dom ∅(𝐹‘(∅‘x)) V)
19 iunexg 5665 . . . . . 6 ((dom ∅ V x dom ∅(𝐹‘(∅‘x)) V) → x dom ∅(𝐹‘(∅‘x)) V)
2011, 18, 19sylancr 395 . . . . 5 (𝐹 Fn V → x dom ∅(𝐹‘(∅‘x)) V)
21 unexg 4124 . . . . . 6 ((A 𝑉 x dom ∅(𝐹‘(∅‘x)) V) → (A x dom ∅(𝐹‘(∅‘x))) V)
2221ex 108 . . . . 5 (A 𝑉 → ( x dom ∅(𝐹‘(∅‘x)) V → (A x dom ∅(𝐹‘(∅‘x))) V))
2320, 22mpan9 265 . . . 4 ((𝐹 Fn V A 𝑉) → (A x dom ∅(𝐹‘(∅‘x))) V)
24 dmeq 4458 . . . . . . 7 (g = ∅ → dom g = dom ∅)
25 fveq1 5098 . . . . . . . 8 (g = ∅ → (gx) = (∅‘x))
2625fveq2d 5103 . . . . . . 7 (g = ∅ → (𝐹‘(gx)) = (𝐹‘(∅‘x)))
2724, 26iuneq12d 3651 . . . . . 6 (g = ∅ → x dom g(𝐹‘(gx)) = x dom ∅(𝐹‘(∅‘x)))
2827uneq2d 3070 . . . . 5 (g = ∅ → (A x dom g(𝐹‘(gx))) = (A x dom ∅(𝐹‘(∅‘x))))
29 eqid 2018 . . . . 5 (g V ↦ (A x dom g(𝐹‘(gx)))) = (g V ↦ (A x dom g(𝐹‘(gx))))
3028, 29fvmptg 5169 . . . 4 ((∅ V (A x dom ∅(𝐹‘(∅‘x))) V) → ((g V ↦ (A x dom g(𝐹‘(gx))))‘∅) = (A x dom ∅(𝐹‘(∅‘x))))
3110, 23, 30sylancr 395 . . 3 ((𝐹 Fn V A 𝑉) → ((g V ↦ (A x dom g(𝐹‘(gx))))‘∅) = (A x dom ∅(𝐹‘(∅‘x))))
329, 31eqtrd 2050 . 2 ((𝐹 Fn V A 𝑉) → (rec(𝐹, A)‘∅) = (A x dom ∅(𝐹‘(∅‘x))))
33 dm0 4472 . . . . . 6 dom ∅ = ∅
34 iuneq1 3640 . . . . . 6 (dom ∅ = ∅ → x dom ∅(𝐹‘(∅‘x)) = x ∅ (𝐹‘(∅‘x)))
3533, 34ax-mp 7 . . . . 5 x dom ∅(𝐹‘(∅‘x)) = x ∅ (𝐹‘(∅‘x))
36 0iun 3684 . . . . 5 x ∅ (𝐹‘(∅‘x)) = ∅
3735, 36eqtri 2038 . . . 4 x dom ∅(𝐹‘(∅‘x)) = ∅
3837uneq2i 3067 . . 3 (A x dom ∅(𝐹‘(∅‘x))) = (A ∪ ∅)
39 un0 3224 . . 3 (A ∪ ∅) = A
4038, 39eqtri 2038 . 2 (A x dom ∅(𝐹‘(∅‘x))) = A
4132, 40syl6eq 2066 1 ((𝐹 Fn V A 𝑉) → (rec(𝐹, A)‘∅) = A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1226   wcel 1370  wral 2280  Vcvv 2531  cun 2888  c0 3197   ciun 3627  cmpt 3788  Oncon0 4045  dom cdm 4268  cres 4270  Fun wfun 4819   Fn wfn 4820  cfv 4825  reccrdg 5873
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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-coll 3842  ax-sep 3845  ax-nul 3853  ax-pow 3897  ax-pr 3914  ax-un 4116  ax-setind 4200
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-ral 2285  df-rex 2286  df-reu 2287  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-nul 3198  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-tr 3825  df-id 4000  df-iord 4048  df-on 4050  df-suc 4053  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832  df-fv 4833  df-recs 5838  df-irdg 5874
This theorem is referenced by:  frecrdg  5904  oa0  5948  om0  5949  oei0  5950
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