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Theorem rdgivallem 6051
Description: Value of the recursive definition generator. Lemma for rdgival 6052 which simplifies the value further. (Contributed by Jim Kingdon, 13-Jul-2019.) (New usage is discouraged.)
Assertion
Ref Expression
rdgivallem ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) = (𝐴 𝑥𝐵 (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 6040 . . . 4 rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))))
2 rdgruledefgg 6045 . . . . 5 ((𝐹 Fn V ∧ 𝐴𝑉) → (Fun (𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))) ∧ ((𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥))))‘𝑦) ∈ V))
32alrimiv 1797 . . . 4 ((𝐹 Fn V ∧ 𝐴𝑉) → ∀𝑦(Fun (𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))) ∧ ((𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥))))‘𝑦) ∈ V))
41, 3tfri2d 6006 . . 3 (((𝐹 Fn V ∧ 𝐴𝑉) ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) = ((𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥))))‘(rec(𝐹, 𝐴) ↾ 𝐵)))
543impa 1134 . 2 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) = ((𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥))))‘(rec(𝐹, 𝐴) ↾ 𝐵)))
6 eqidd 2084 . . 3 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → (𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))) = (𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))))
7 dmeq 4583 . . . . . 6 (𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵) → dom 𝑔 = dom (rec(𝐹, 𝐴) ↾ 𝐵))
8 onss 4265 . . . . . . . . 9 (𝐵 ∈ On → 𝐵 ⊆ On)
983ad2ant3 962 . . . . . . . 8 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → 𝐵 ⊆ On)
10 rdgifnon 6049 . . . . . . . . . 10 ((𝐹 Fn V ∧ 𝐴𝑉) → rec(𝐹, 𝐴) Fn On)
11 fndm 5049 . . . . . . . . . 10 (rec(𝐹, 𝐴) Fn On → dom rec(𝐹, 𝐴) = On)
1210, 11syl 14 . . . . . . . . 9 ((𝐹 Fn V ∧ 𝐴𝑉) → dom rec(𝐹, 𝐴) = On)
13123adant3 959 . . . . . . . 8 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → dom rec(𝐹, 𝐴) = On)
149, 13sseqtr4d 3045 . . . . . . 7 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → 𝐵 ⊆ dom rec(𝐹, 𝐴))
15 ssdmres 4681 . . . . . . 7 (𝐵 ⊆ dom rec(𝐹, 𝐴) ↔ dom (rec(𝐹, 𝐴) ↾ 𝐵) = 𝐵)
1614, 15sylib 120 . . . . . 6 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → dom (rec(𝐹, 𝐴) ↾ 𝐵) = 𝐵)
177, 16sylan9eqr 2137 . . . . 5 (((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) ∧ 𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵)) → dom 𝑔 = 𝐵)
18 fveq1 5229 . . . . . . 7 (𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵) → (𝑔𝑥) = ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))
1918fveq2d 5234 . . . . . 6 (𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵) → (𝐹‘(𝑔𝑥)) = (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)))
2019adantl 271 . . . . 5 (((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) ∧ 𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵)) → (𝐹‘(𝑔𝑥)) = (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)))
2117, 20iuneq12d 3722 . . . 4 (((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) ∧ 𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵)) → 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)) = 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)))
2221uneq2d 3136 . . 3 (((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) ∧ 𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵)) → (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥))) = (𝐴 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))))
23 rdgfun 6043 . . . . 5 Fun rec(𝐹, 𝐴)
24 resfunexg 5435 . . . . 5 ((Fun rec(𝐹, 𝐴) ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴) ↾ 𝐵) ∈ V)
2523, 24mpan 415 . . . 4 (𝐵 ∈ On → (rec(𝐹, 𝐴) ↾ 𝐵) ∈ V)
26253ad2ant3 962 . . 3 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → (rec(𝐹, 𝐴) ↾ 𝐵) ∈ V)
27 simpr 108 . . . . . 6 ((𝐹 Fn V ∧ 𝐵 ∈ On) → 𝐵 ∈ On)
28 vex 2613 . . . . . . . . . 10 𝑥 ∈ V
29 fvexg 5246 . . . . . . . . . 10 (((rec(𝐹, 𝐴) ↾ 𝐵) ∈ V ∧ 𝑥 ∈ V) → ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V)
3025, 28, 29sylancl 404 . . . . . . . . 9 (𝐵 ∈ On → ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V)
3130ralrimivw 2440 . . . . . . . 8 (𝐵 ∈ On → ∀𝑥𝐵 ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V)
3231adantl 271 . . . . . . 7 ((𝐹 Fn V ∧ 𝐵 ∈ On) → ∀𝑥𝐵 ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V)
33 funfvex 5244 . . . . . . . . . . 11 ((Fun 𝐹 ∧ ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ dom 𝐹) → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
3433funfni 5050 . . . . . . . . . 10 ((𝐹 Fn V ∧ ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V) → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
3534ex 113 . . . . . . . . 9 (𝐹 Fn V → (((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V))
3635ralimdv 2435 . . . . . . . 8 (𝐹 Fn V → (∀𝑥𝐵 ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V → ∀𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V))
3736adantr 270 . . . . . . 7 ((𝐹 Fn V ∧ 𝐵 ∈ On) → (∀𝑥𝐵 ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V → ∀𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V))
3832, 37mpd 13 . . . . . 6 ((𝐹 Fn V ∧ 𝐵 ∈ On) → ∀𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
39 iunexg 5798 . . . . . 6 ((𝐵 ∈ On ∧ ∀𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V) → 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
4027, 38, 39syl2anc 403 . . . . 5 ((𝐹 Fn V ∧ 𝐵 ∈ On) → 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
41403adant2 958 . . . 4 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
42 unexg 4224 . . . . . 6 ((𝐴𝑉 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V) → (𝐴 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))) ∈ V)
4342ex 113 . . . . 5 (𝐴𝑉 → ( 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V → (𝐴 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))) ∈ V))
44433ad2ant2 961 . . . 4 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → ( 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V → (𝐴 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))) ∈ V))
4541, 44mpd 13 . . 3 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → (𝐴 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))) ∈ V)
466, 22, 26, 45fvmptd 5306 . 2 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → ((𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥))))‘(rec(𝐹, 𝐴) ↾ 𝐵)) = (𝐴 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))))
475, 46eqtrd 2115 1 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) = (𝐴 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 920   = wceq 1285  wcel 1434  wral 2353  Vcvv 2610  cun 2980  wss 2982   ciun 3698  cmpt 3859  Oncon0 4146  dom cdm 4391  cres 4393  Fun wfun 4946   Fn wfn 4947  cfv 4952  reccrdg 6039
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-suc 4154  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-recs 5975  df-irdg 6040
This theorem is referenced by:  rdgival  6052
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