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Theorem rdgivallem 6382
Description: Value of the recursive definition generator. Lemma for rdgival 6383 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 6371 . . . 4 rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))))
2 rdgruledefgg 6376 . . . . 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 1874 . . . 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 6337 . . 3 (((𝐹 Fn V ∧ 𝐴𝑉) ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) = ((𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥))))‘(rec(𝐹, 𝐴) ↾ 𝐵)))
543impa 1194 . 2 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) = ((𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥))))‘(rec(𝐹, 𝐴) ↾ 𝐵)))
6 eqidd 2178 . . 3 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → (𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))) = (𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))))
7 dmeq 4828 . . . . . 6 (𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵) → dom 𝑔 = dom (rec(𝐹, 𝐴) ↾ 𝐵))
8 onss 4493 . . . . . . . . 9 (𝐵 ∈ On → 𝐵 ⊆ On)
983ad2ant3 1020 . . . . . . . 8 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → 𝐵 ⊆ On)
10 rdgifnon 6380 . . . . . . . . . 10 ((𝐹 Fn V ∧ 𝐴𝑉) → rec(𝐹, 𝐴) Fn On)
11 fndm 5316 . . . . . . . . . 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 1017 . . . . . . . 8 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → dom rec(𝐹, 𝐴) = On)
149, 13sseqtrrd 3195 . . . . . . 7 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → 𝐵 ⊆ dom rec(𝐹, 𝐴))
15 ssdmres 4930 . . . . . . 7 (𝐵 ⊆ dom rec(𝐹, 𝐴) ↔ dom (rec(𝐹, 𝐴) ↾ 𝐵) = 𝐵)
1614, 15sylib 122 . . . . . 6 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → dom (rec(𝐹, 𝐴) ↾ 𝐵) = 𝐵)
177, 16sylan9eqr 2232 . . . . 5 (((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) ∧ 𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵)) → dom 𝑔 = 𝐵)
18 fveq1 5515 . . . . . . 7 (𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵) → (𝑔𝑥) = ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))
1918fveq2d 5520 . . . . . 6 (𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵) → (𝐹‘(𝑔𝑥)) = (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)))
2019adantl 277 . . . . 5 (((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) ∧ 𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵)) → (𝐹‘(𝑔𝑥)) = (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)))
2117, 20iuneq12d 3911 . . . 4 (((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) ∧ 𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵)) → 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)) = 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)))
2221uneq2d 3290 . . 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 6374 . . . . 5 Fun rec(𝐹, 𝐴)
24 resfunexg 5738 . . . . 5 ((Fun rec(𝐹, 𝐴) ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴) ↾ 𝐵) ∈ V)
2523, 24mpan 424 . . . 4 (𝐵 ∈ On → (rec(𝐹, 𝐴) ↾ 𝐵) ∈ V)
26253ad2ant3 1020 . . 3 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → (rec(𝐹, 𝐴) ↾ 𝐵) ∈ V)
27 simpr 110 . . . . . 6 ((𝐹 Fn V ∧ 𝐵 ∈ On) → 𝐵 ∈ On)
28 vex 2741 . . . . . . . . . 10 𝑥 ∈ V
29 fvexg 5535 . . . . . . . . . 10 (((rec(𝐹, 𝐴) ↾ 𝐵) ∈ V ∧ 𝑥 ∈ V) → ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V)
3025, 28, 29sylancl 413 . . . . . . . . 9 (𝐵 ∈ On → ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V)
3130ralrimivw 2551 . . . . . . . 8 (𝐵 ∈ On → ∀𝑥𝐵 ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V)
3231adantl 277 . . . . . . 7 ((𝐹 Fn V ∧ 𝐵 ∈ On) → ∀𝑥𝐵 ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V)
33 funfvex 5533 . . . . . . . . . . 11 ((Fun 𝐹 ∧ ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ dom 𝐹) → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
3433funfni 5317 . . . . . . . . . 10 ((𝐹 Fn V ∧ ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V) → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
3534ex 115 . . . . . . . . 9 (𝐹 Fn V → (((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V))
3635ralimdv 2545 . . . . . . . 8 (𝐹 Fn V → (∀𝑥𝐵 ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V → ∀𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V))
3736adantr 276 . . . . . . 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 6120 . . . . . 6 ((𝐵 ∈ On ∧ ∀𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V) → 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
4027, 38, 39syl2anc 411 . . . . 5 ((𝐹 Fn V ∧ 𝐵 ∈ On) → 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
41403adant2 1016 . . . 4 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
42 unexg 4444 . . . . . 6 ((𝐴𝑉 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V) → (𝐴 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))) ∈ V)
4342ex 115 . . . . 5 (𝐴𝑉 → ( 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V → (𝐴 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))) ∈ V))
44433ad2ant2 1019 . . . 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 5598 . 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 2210 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 104  w3a 978   = wceq 1353  wcel 2148  wral 2455  Vcvv 2738  cun 3128  wss 3130   ciun 3887  cmpt 4065  Oncon0 4364  dom cdm 4627  cres 4629  Fun wfun 5211   Fn wfn 5212  cfv 5217  reccrdg 6370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-recs 6306  df-irdg 6371
This theorem is referenced by:  rdgival  6383
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