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Theorem rdgivallem 5998
Description: Value of the recursive definition generator. Lemma for rdgival 5999 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 5987 . . . 4 rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))))
2 rdgruledefgg 5992 . . . . 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 1770 . . . 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 5980 . . 3 (((𝐹 Fn V ∧ 𝐴𝑉) ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) = ((𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥))))‘(rec(𝐹, 𝐴) ↾ 𝐵)))
543impa 1110 . 2 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) = ((𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥))))‘(rec(𝐹, 𝐴) ↾ 𝐵)))
6 eqidd 2057 . . 3 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → (𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))) = (𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))))
7 dmeq 4562 . . . . . 6 (𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵) → dom 𝑔 = dom (rec(𝐹, 𝐴) ↾ 𝐵))
8 onss 4246 . . . . . . . . 9 (𝐵 ∈ On → 𝐵 ⊆ On)
983ad2ant3 938 . . . . . . . 8 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → 𝐵 ⊆ On)
10 rdgifnon 5996 . . . . . . . . . 10 ((𝐹 Fn V ∧ 𝐴𝑉) → rec(𝐹, 𝐴) Fn On)
11 fndm 5025 . . . . . . . . . 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 935 . . . . . . . 8 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → dom rec(𝐹, 𝐴) = On)
149, 13sseqtr4d 3009 . . . . . . 7 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → 𝐵 ⊆ dom rec(𝐹, 𝐴))
15 ssdmres 4660 . . . . . . 7 (𝐵 ⊆ dom rec(𝐹, 𝐴) ↔ dom (rec(𝐹, 𝐴) ↾ 𝐵) = 𝐵)
1614, 15sylib 131 . . . . . 6 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → dom (rec(𝐹, 𝐴) ↾ 𝐵) = 𝐵)
177, 16sylan9eqr 2110 . . . . 5 (((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) ∧ 𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵)) → dom 𝑔 = 𝐵)
18 fveq1 5204 . . . . . . 7 (𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵) → (𝑔𝑥) = ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))
1918fveq2d 5209 . . . . . 6 (𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵) → (𝐹‘(𝑔𝑥)) = (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)))
2019adantl 266 . . . . 5 (((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) ∧ 𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵)) → (𝐹‘(𝑔𝑥)) = (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)))
2117, 20iuneq12d 3708 . . . 4 (((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) ∧ 𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵)) → 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)) = 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)))
2221uneq2d 3124 . . 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 5990 . . . . 5 Fun rec(𝐹, 𝐴)
24 resfunexg 5409 . . . . 5 ((Fun rec(𝐹, 𝐴) ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴) ↾ 𝐵) ∈ V)
2523, 24mpan 408 . . . 4 (𝐵 ∈ On → (rec(𝐹, 𝐴) ↾ 𝐵) ∈ V)
26253ad2ant3 938 . . 3 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → (rec(𝐹, 𝐴) ↾ 𝐵) ∈ V)
27 simpr 107 . . . . . 6 ((𝐹 Fn V ∧ 𝐵 ∈ On) → 𝐵 ∈ On)
28 vex 2577 . . . . . . . . . 10 𝑥 ∈ V
29 fvexg 5221 . . . . . . . . . 10 (((rec(𝐹, 𝐴) ↾ 𝐵) ∈ V ∧ 𝑥 ∈ V) → ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V)
3025, 28, 29sylancl 398 . . . . . . . . 9 (𝐵 ∈ On → ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V)
3130ralrimivw 2410 . . . . . . . 8 (𝐵 ∈ On → ∀𝑥𝐵 ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V)
3231adantl 266 . . . . . . 7 ((𝐹 Fn V ∧ 𝐵 ∈ On) → ∀𝑥𝐵 ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V)
33 funfvex 5219 . . . . . . . . . . 11 ((Fun 𝐹 ∧ ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ dom 𝐹) → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
3433funfni 5026 . . . . . . . . . 10 ((𝐹 Fn V ∧ ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V) → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
3534ex 112 . . . . . . . . 9 (𝐹 Fn V → (((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V))
3635ralimdv 2405 . . . . . . . 8 (𝐹 Fn V → (∀𝑥𝐵 ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V → ∀𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V))
3736adantr 265 . . . . . . 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 5773 . . . . . 6 ((𝐵 ∈ On ∧ ∀𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V) → 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
4027, 38, 39syl2anc 397 . . . . 5 ((𝐹 Fn V ∧ 𝐵 ∈ On) → 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
41403adant2 934 . . . 4 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
42 unexg 4205 . . . . . 6 ((𝐴𝑉 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V) → (𝐴 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))) ∈ V)
4342ex 112 . . . . 5 (𝐴𝑉 → ( 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V → (𝐴 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))) ∈ V))
44433ad2ant2 937 . . . 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 5280 . 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 2088 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 101  w3a 896   = wceq 1259  wcel 1409  wral 2323  Vcvv 2574  cun 2942  wss 2944   ciun 3684  cmpt 3845  Oncon0 4127  dom cdm 4372  cres 4374  Fun wfun 4923   Fn wfn 4924  cfv 4929  reccrdg 5986
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-id 4057  df-iord 4130  df-on 4132  df-suc 4135  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-recs 5950  df-irdg 5987
This theorem is referenced by:  rdgival  5999  rdgon  6003
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