ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rdgivallem Structured version   GIF version

Theorem rdgivallem 5908
Description: Value of the recursive definition generator. Lemma for rdgival 5909 which simplifies the value further. (Contributed by Jim Kingdon, 13-Jul-2019.) (New usage is discouraged.)
Assertion
Ref Expression
rdgivallem ((𝐹 Fn V A 𝑉 B On) → (rec(𝐹, A)‘B) = (A x B (𝐹‘((rec(𝐹, A) ↾ B)‘x))))
Distinct variable groups:   x,A   x,B   x,𝐹   x,𝑉

Proof of Theorem rdgivallem
Dummy variables g y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-irdg 5897 . . . 4 rec(𝐹, A) = recs((g V ↦ (A x dom g(𝐹‘(gx)))))
2 rdgruledefgg 5902 . . . . 5 ((𝐹 Fn V A 𝑉) → (Fun (g V ↦ (A x dom g(𝐹‘(gx)))) ((g V ↦ (A x dom g(𝐹‘(gx))))‘y) V))
32alrimiv 1751 . . . 4 ((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))
41, 3tfri2d 5891 . . 3 (((𝐹 Fn V A 𝑉) B On) → (rec(𝐹, A)‘B) = ((g V ↦ (A x dom g(𝐹‘(gx))))‘(rec(𝐹, A) ↾ B)))
543impa 1098 . 2 ((𝐹 Fn V A 𝑉 B On) → (rec(𝐹, A)‘B) = ((g V ↦ (A x dom g(𝐹‘(gx))))‘(rec(𝐹, A) ↾ B)))
6 eqidd 2038 . . 3 ((𝐹 Fn V A 𝑉 B On) → (g V ↦ (A x dom g(𝐹‘(gx)))) = (g V ↦ (A x dom g(𝐹‘(gx)))))
7 dmeq 4478 . . . . . 6 (g = (rec(𝐹, A) ↾ B) → dom g = dom (rec(𝐹, A) ↾ B))
8 onss 4185 . . . . . . . . 9 (B On → B ⊆ On)
983ad2ant3 926 . . . . . . . 8 ((𝐹 Fn V A 𝑉 B On) → B ⊆ On)
10 rdgifnon 5906 . . . . . . . . . 10 ((𝐹 Fn V A 𝑉) → rec(𝐹, A) Fn On)
11 fndm 4941 . . . . . . . . . 10 (rec(𝐹, A) Fn On → dom rec(𝐹, A) = On)
1210, 11syl 14 . . . . . . . . 9 ((𝐹 Fn V A 𝑉) → dom rec(𝐹, A) = On)
13123adant3 923 . . . . . . . 8 ((𝐹 Fn V A 𝑉 B On) → dom rec(𝐹, A) = On)
149, 13sseqtr4d 2976 . . . . . . 7 ((𝐹 Fn V A 𝑉 B On) → B ⊆ dom rec(𝐹, A))
15 ssdmres 4576 . . . . . . 7 (B ⊆ dom rec(𝐹, A) ↔ dom (rec(𝐹, A) ↾ B) = B)
1614, 15sylib 127 . . . . . 6 ((𝐹 Fn V A 𝑉 B On) → dom (rec(𝐹, A) ↾ B) = B)
177, 16sylan9eqr 2091 . . . . 5 (((𝐹 Fn V A 𝑉 B On) g = (rec(𝐹, A) ↾ B)) → dom g = B)
18 fveq1 5120 . . . . . . 7 (g = (rec(𝐹, A) ↾ B) → (gx) = ((rec(𝐹, A) ↾ B)‘x))
1918fveq2d 5125 . . . . . 6 (g = (rec(𝐹, A) ↾ B) → (𝐹‘(gx)) = (𝐹‘((rec(𝐹, A) ↾ B)‘x)))
2019adantl 262 . . . . 5 (((𝐹 Fn V A 𝑉 B On) g = (rec(𝐹, A) ↾ B)) → (𝐹‘(gx)) = (𝐹‘((rec(𝐹, A) ↾ B)‘x)))
2117, 20iuneq12d 3672 . . . 4 (((𝐹 Fn V A 𝑉 B On) g = (rec(𝐹, A) ↾ B)) → x dom g(𝐹‘(gx)) = x B (𝐹‘((rec(𝐹, A) ↾ B)‘x)))
2221uneq2d 3091 . . 3 (((𝐹 Fn V A 𝑉 B On) g = (rec(𝐹, A) ↾ B)) → (A x dom g(𝐹‘(gx))) = (A x B (𝐹‘((rec(𝐹, A) ↾ B)‘x))))
23 rdgfun 5900 . . . . 5 Fun rec(𝐹, A)
24 resfunexg 5325 . . . . 5 ((Fun rec(𝐹, A) B On) → (rec(𝐹, A) ↾ B) V)
2523, 24mpan 400 . . . 4 (B On → (rec(𝐹, A) ↾ B) V)
26253ad2ant3 926 . . 3 ((𝐹 Fn V A 𝑉 B On) → (rec(𝐹, A) ↾ B) V)
27 simpr 103 . . . . . 6 ((𝐹 Fn V B On) → B On)
28 vex 2554 . . . . . . . . . 10 x V
29 fvexg 5137 . . . . . . . . . 10 (((rec(𝐹, A) ↾ B) V x V) → ((rec(𝐹, A) ↾ B)‘x) V)
3025, 28, 29sylancl 392 . . . . . . . . 9 (B On → ((rec(𝐹, A) ↾ B)‘x) V)
3130ralrimivw 2387 . . . . . . . 8 (B On → x B ((rec(𝐹, A) ↾ B)‘x) V)
3231adantl 262 . . . . . . 7 ((𝐹 Fn V B On) → x B ((rec(𝐹, A) ↾ B)‘x) V)
33 funfvex 5135 . . . . . . . . . . 11 ((Fun 𝐹 ((rec(𝐹, A) ↾ B)‘x) dom 𝐹) → (𝐹‘((rec(𝐹, A) ↾ B)‘x)) V)
3433funfni 4942 . . . . . . . . . 10 ((𝐹 Fn V ((rec(𝐹, A) ↾ B)‘x) V) → (𝐹‘((rec(𝐹, A) ↾ B)‘x)) V)
3534ex 108 . . . . . . . . 9 (𝐹 Fn V → (((rec(𝐹, A) ↾ B)‘x) V → (𝐹‘((rec(𝐹, A) ↾ B)‘x)) V))
3635ralimdv 2382 . . . . . . . 8 (𝐹 Fn V → (x B ((rec(𝐹, A) ↾ B)‘x) V → x B (𝐹‘((rec(𝐹, A) ↾ B)‘x)) V))
3736adantr 261 . . . . . . 7 ((𝐹 Fn V B On) → (x B ((rec(𝐹, A) ↾ B)‘x) V → x B (𝐹‘((rec(𝐹, A) ↾ B)‘x)) V))
3832, 37mpd 13 . . . . . 6 ((𝐹 Fn V B On) → x B (𝐹‘((rec(𝐹, A) ↾ B)‘x)) V)
39 iunexg 5688 . . . . . 6 ((B On x B (𝐹‘((rec(𝐹, A) ↾ B)‘x)) V) → x B (𝐹‘((rec(𝐹, A) ↾ B)‘x)) V)
4027, 38, 39syl2anc 391 . . . . 5 ((𝐹 Fn V B On) → x B (𝐹‘((rec(𝐹, A) ↾ B)‘x)) V)
41403adant2 922 . . . 4 ((𝐹 Fn V A 𝑉 B On) → x B (𝐹‘((rec(𝐹, A) ↾ B)‘x)) V)
42 unexg 4144 . . . . . 6 ((A 𝑉 x B (𝐹‘((rec(𝐹, A) ↾ B)‘x)) V) → (A x B (𝐹‘((rec(𝐹, A) ↾ B)‘x))) V)
4342ex 108 . . . . 5 (A 𝑉 → ( x B (𝐹‘((rec(𝐹, A) ↾ B)‘x)) V → (A x B (𝐹‘((rec(𝐹, A) ↾ B)‘x))) V))
44433ad2ant2 925 . . . 4 ((𝐹 Fn V A 𝑉 B On) → ( x B (𝐹‘((rec(𝐹, A) ↾ B)‘x)) V → (A x B (𝐹‘((rec(𝐹, A) ↾ B)‘x))) V))
4541, 44mpd 13 . . 3 ((𝐹 Fn V A 𝑉 B On) → (A x B (𝐹‘((rec(𝐹, A) ↾ B)‘x))) V)
466, 22, 26, 45fvmptd 5196 . 2 ((𝐹 Fn V A 𝑉 B On) → ((g V ↦ (A x dom g(𝐹‘(gx))))‘(rec(𝐹, A) ↾ B)) = (A x B (𝐹‘((rec(𝐹, A) ↾ B)‘x))))
475, 46eqtrd 2069 1 ((𝐹 Fn V A 𝑉 B On) → (rec(𝐹, A)‘B) = (A x B (𝐹‘((rec(𝐹, A) ↾ B)‘x))))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884   = wceq 1242   wcel 1390  wral 2300  Vcvv 2551  cun 2909  wss 2911   ciun 3648  cmpt 3809  Oncon0 4066  dom cdm 4288  cres 4290  Fun wfun 4839   Fn wfn 4840  cfv 4845  reccrdg 5896
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 544  ax-in2 545  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  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  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-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  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-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  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  df-recs 5861  df-irdg 5897
This theorem is referenced by:  rdgival  5909  rdgon  5913
  Copyright terms: Public domain W3C validator