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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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