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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
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