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Theorem rdgruledefgg 5902
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 2560 . 2 (A 𝑉A V)
2 funmpt 4881 . . . 4 Fun (g V ↦ (A x dom g(𝐹‘(gx))))
3 vex 2554 . . . . 5 f V
4 vex 2554 . . . . . . . . . . . . 13 g V
5 vex 2554 . . . . . . . . . . . . 13 x V
64, 5fvex 5138 . . . . . . . . . . . 12 (gx) V
7 funfvex 5135 . . . . . . . . . . . . 13 ((Fun 𝐹 (gx) dom 𝐹) → (𝐹‘(gx)) V)
87funfni 4942 . . . . . . . . . . . 12 ((𝐹 Fn V (gx) V) → (𝐹‘(gx)) V)
96, 8mpan2 401 . . . . . . . . . . 11 (𝐹 Fn V → (𝐹‘(gx)) V)
109ralrimivw 2387 . . . . . . . . . 10 (𝐹 Fn V → x dom g(𝐹‘(gx)) V)
114dmex 4541 . . . . . . . . . . 11 dom g V
12 iunexg 5688 . . . . . . . . . . 11 ((dom g V x dom g(𝐹‘(gx)) V) → x dom g(𝐹‘(gx)) V)
1311, 12mpan 400 . . . . . . . . . 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 4144 . . . . . . . . 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 2387 . . . . . 6 ((𝐹 Fn V A V) → g V (A x dom g(𝐹‘(gx))) V)
19 dmmptg 4761 . . . . . 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 2124 . . . 4 ((𝐹 Fn V A V) → f dom (g V ↦ (A x dom g(𝐹‘(gx)))))
22 funfvex 5135 . . . 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 393 . . 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 1242   wcel 1390  wral 2300  Vcvv 2551  cun 2909   ciun 3648  cmpt 3809  dom cdm 4288  Fun wfun 4839   Fn wfn 4840  cfv 4845
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853
This theorem is referenced by:  rdgruledefg  5903  rdgexggg  5904  rdgifnon  5906  rdgivallem  5908
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