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

Theorem rdgivallem 5884
Description: Value of the recursive definition generator. Lemma for rdgival 5885 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 5874 . . . 4 rec(𝐹, A) = recs((g V ↦ (A x dom g(𝐹‘(gx)))))
2 rdgruledefgg 5878 . . . . 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 1732 . . . 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 5868 . . 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 1083 . 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 2019 . . 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 4458 . . . . . 6 (g = (rec(𝐹, A) ↾ B) → dom g = dom (rec(𝐹, A) ↾ B))
8 onss 4165 . . . . . . . . 9 (B On → B ⊆ On)
983ad2ant3 913 . . . . . . . 8 ((𝐹 Fn V A 𝑉 B On) → B ⊆ On)
10 rdgifnon 5883 . . . . . . . . . 10 ((𝐹 Fn V A 𝑉) → rec(𝐹, A) Fn On)
11 fndm 4920 . . . . . . . . . 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 910 . . . . . . . 8 ((𝐹 Fn V A 𝑉 B On) → dom rec(𝐹, A) = On)
149, 13sseqtr4d 2955 . . . . . . 7 ((𝐹 Fn V A 𝑉 B On) → B ⊆ dom rec(𝐹, A))
15 ssdmres 4556 . . . . . . 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 2072 . . . . 5 (((𝐹 Fn V A 𝑉 B On) g = (rec(𝐹, A) ↾ B)) → dom g = B)
18 fveq1 5098 . . . . . . 7 (g = (rec(𝐹, A) ↾ B) → (gx) = ((rec(𝐹, A) ↾ B)‘x))
1918fveq2d 5103 . . . . . 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 3651 . . . 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 3070 . . 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 5877 . . . . 5 Fun rec(𝐹, A)
24 resfunexg 5303 . . . . 5 ((Fun rec(𝐹, A) B On) → (rec(𝐹, A) ↾ B) V)
2523, 24mpan 402 . . . 4 (B On → (rec(𝐹, A) ↾ B) V)
26253ad2ant3 913 . . 3 ((𝐹 Fn V A 𝑉 B On) → (rec(𝐹, A) ↾ B) V)
27 ax-ia2 100 . . . . . 6 ((𝐹 Fn V B On) → B On)
28 vex 2534 . . . . . . . . . 10 x V
29 fvexg 5115 . . . . . . . . . 10 (((rec(𝐹, A) ↾ B) V x V) → ((rec(𝐹, A) ↾ B)‘x) V)
3025, 28, 29sylancl 394 . . . . . . . . 9 (B On → ((rec(𝐹, A) ↾ B)‘x) V)
3130ralrimivw 2367 . . . . . . . 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 5113 . . . . . . . . . . 11 ((Fun 𝐹 ((rec(𝐹, A) ↾ B)‘x) dom 𝐹) → (𝐹‘((rec(𝐹, A) ↾ B)‘x)) V)
3433funfni 4921 . . . . . . . . . 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 2362 . . . . . . . 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 5665 . . . . . 6 ((B On x B (𝐹‘((rec(𝐹, A) ↾ B)‘x)) V) → x B (𝐹‘((rec(𝐹, A) ↾ B)‘x)) V)
4027, 38, 39syl2anc 393 . . . . 5 ((𝐹 Fn V B On) → x B (𝐹‘((rec(𝐹, A) ↾ B)‘x)) V)
41403adant2 909 . . . 4 ((𝐹 Fn V A 𝑉 B On) → x B (𝐹‘((rec(𝐹, A) ↾ B)‘x)) V)
42 unexg 4124 . . . . . 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 912 . . . 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 5174 . 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 2050 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 871   = wceq 1226   wcel 1370  wral 2280  Vcvv 2531  cun 2888  wss 2890   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-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:  rdgival  5885  rdgon  5889
  Copyright terms: Public domain W3C validator