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Theorem rdgruledefgg 5882
 Description: The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.)
Assertion
Ref Expression
rdgruledefgg ((𝐹 Fn V A 𝑉) → (Fun (g V ↦ (A x dom g(𝐹‘(gx)))) ((g V ↦ (A x dom g(𝐹‘(gx))))‘f) V))
Distinct variable groups:   A,g   x,g,𝐹
Allowed substitution hints:   A(x,f)   𝐹(f)   𝑉(x,f,g)

Proof of Theorem rdgruledefgg
StepHypRef Expression
1 elex 2543 . 2 (A 𝑉A V)
2 funmpt 4864 . . . 4 Fun (g V ↦ (A x dom g(𝐹‘(gx))))
3 vex 2538 . . . . 5 f V
4 vex 2538 . . . . . . . . . . . . 13 g V
5 vex 2538 . . . . . . . . . . . . 13 x V
64, 5fvex 5120 . . . . . . . . . . . 12 (gx) V
7 funfvex 5117 . . . . . . . . . . . . 13 ((Fun 𝐹 (gx) dom 𝐹) → (𝐹‘(gx)) V)
87funfni 4925 . . . . . . . . . . . 12 ((𝐹 Fn V (gx) V) → (𝐹‘(gx)) V)
96, 8mpan2 403 . . . . . . . . . . 11 (𝐹 Fn V → (𝐹‘(gx)) V)
109ralrimivw 2371 . . . . . . . . . 10 (𝐹 Fn V → x dom g(𝐹‘(gx)) V)
114dmex 4525 . . . . . . . . . . 11 dom g V
12 iunexg 5669 . . . . . . . . . . 11 ((dom g V x dom g(𝐹‘(gx)) V) → x dom g(𝐹‘(gx)) V)
1311, 12mpan 402 . . . . . . . . . 10 (x dom g(𝐹‘(gx)) V → x dom g(𝐹‘(gx)) V)
1410, 13syl 14 . . . . . . . . 9 (𝐹 Fn V → x dom g(𝐹‘(gx)) V)
15 unexg 4128 . . . . . . . . 9 ((A V x dom g(𝐹‘(gx)) V) → (A x dom g(𝐹‘(gx))) V)
1614, 15sylan2 270 . . . . . . . 8 ((A V 𝐹 Fn V) → (A x dom g(𝐹‘(gx))) V)
1716ancoms 255 . . . . . . 7 ((𝐹 Fn V A V) → (A x dom g(𝐹‘(gx))) V)
1817ralrimivw 2371 . . . . . 6 ((𝐹 Fn V A V) → g V (A x dom g(𝐹‘(gx))) V)
19 dmmptg 4745 . . . . . 6 (g V (A x dom g(𝐹‘(gx))) V → dom (g V ↦ (A x dom g(𝐹‘(gx)))) = V)
2018, 19syl 14 . . . . 5 ((𝐹 Fn V A V) → dom (g V ↦ (A x dom g(𝐹‘(gx)))) = V)
213, 20syl5eleqr 2109 . . . 4 ((𝐹 Fn V A V) → f dom (g V ↦ (A x dom g(𝐹‘(gx)))))
22 funfvex 5117 . . . 4 ((Fun (g V ↦ (A x dom g(𝐹‘(gx)))) f dom (g V ↦ (A x dom g(𝐹‘(gx))))) → ((g V ↦ (A x dom g(𝐹‘(gx))))‘f) V)
232, 21, 22sylancr 395 . . 3 ((𝐹 Fn V A V) → ((g V ↦ (A x dom g(𝐹‘(gx))))‘f) V)
2423, 2jctil 295 . 2 ((𝐹 Fn V A V) → (Fun (g V ↦ (A x dom g(𝐹‘(gx)))) ((g V ↦ (A x dom g(𝐹‘(gx))))‘f) V))
251, 24sylan2 270 1 ((𝐹 Fn V A 𝑉) → (Fun (g V ↦ (A x dom g(𝐹‘(gx)))) ((g V ↦ (A x dom g(𝐹‘(gx))))‘f) V))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1228   ∈ wcel 1374  ∀wral 2284  Vcvv 2535   ∪ cun 2892  ∪ ciun 3631   ↦ cmpt 3792  dom cdm 4272  Fun wfun 4823   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 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-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  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:  rdgruledefg  5883  rdgexggg  5884  rdgi0g  5886  rdgifnon  5887  rdgivallem  5888  rdgruledef  5894
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