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Theorem rdgivallem 5968
Description: Value of the recursive definition generator. Lemma for rdgival 5969 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 5957 . . . 4 rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))))
2 rdgruledefgg 5962 . . . . 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 1754 . . . 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 5950 . . 3 (((𝐹 Fn V ∧ 𝐴𝑉) ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) = ((𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥))))‘(rec(𝐹, 𝐴) ↾ 𝐵)))
543impa 1099 . 2 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) = ((𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥))))‘(rec(𝐹, 𝐴) ↾ 𝐵)))
6 eqidd 2041 . . 3 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → (𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))) = (𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))))
7 dmeq 4535 . . . . . 6 (𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵) → dom 𝑔 = dom (rec(𝐹, 𝐴) ↾ 𝐵))
8 onss 4219 . . . . . . . . 9 (𝐵 ∈ On → 𝐵 ⊆ On)
983ad2ant3 927 . . . . . . . 8 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → 𝐵 ⊆ On)
10 rdgifnon 5966 . . . . . . . . . 10 ((𝐹 Fn V ∧ 𝐴𝑉) → rec(𝐹, 𝐴) Fn On)
11 fndm 4998 . . . . . . . . . 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 924 . . . . . . . 8 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → dom rec(𝐹, 𝐴) = On)
149, 13sseqtr4d 2982 . . . . . . 7 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → 𝐵 ⊆ dom rec(𝐹, 𝐴))
15 ssdmres 4633 . . . . . . 7 (𝐵 ⊆ dom rec(𝐹, 𝐴) ↔ dom (rec(𝐹, 𝐴) ↾ 𝐵) = 𝐵)
1614, 15sylib 127 . . . . . 6 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → dom (rec(𝐹, 𝐴) ↾ 𝐵) = 𝐵)
177, 16sylan9eqr 2094 . . . . 5 (((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) ∧ 𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵)) → dom 𝑔 = 𝐵)
18 fveq1 5177 . . . . . . 7 (𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵) → (𝑔𝑥) = ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))
1918fveq2d 5182 . . . . . 6 (𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵) → (𝐹‘(𝑔𝑥)) = (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)))
2019adantl 262 . . . . 5 (((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) ∧ 𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵)) → (𝐹‘(𝑔𝑥)) = (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)))
2117, 20iuneq12d 3681 . . . 4 (((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) ∧ 𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵)) → 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)) = 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)))
2221uneq2d 3097 . . 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 5960 . . . . 5 Fun rec(𝐹, 𝐴)
24 resfunexg 5382 . . . . 5 ((Fun rec(𝐹, 𝐴) ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴) ↾ 𝐵) ∈ V)
2523, 24mpan 400 . . . 4 (𝐵 ∈ On → (rec(𝐹, 𝐴) ↾ 𝐵) ∈ V)
26253ad2ant3 927 . . 3 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → (rec(𝐹, 𝐴) ↾ 𝐵) ∈ V)
27 simpr 103 . . . . . 6 ((𝐹 Fn V ∧ 𝐵 ∈ On) → 𝐵 ∈ On)
28 vex 2560 . . . . . . . . . 10 𝑥 ∈ V
29 fvexg 5194 . . . . . . . . . 10 (((rec(𝐹, 𝐴) ↾ 𝐵) ∈ V ∧ 𝑥 ∈ V) → ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V)
3025, 28, 29sylancl 392 . . . . . . . . 9 (𝐵 ∈ On → ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V)
3130ralrimivw 2393 . . . . . . . 8 (𝐵 ∈ On → ∀𝑥𝐵 ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V)
3231adantl 262 . . . . . . 7 ((𝐹 Fn V ∧ 𝐵 ∈ On) → ∀𝑥𝐵 ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V)
33 funfvex 5192 . . . . . . . . . . 11 ((Fun 𝐹 ∧ ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ dom 𝐹) → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
3433funfni 4999 . . . . . . . . . 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 2388 . . . . . . . 8 (𝐹 Fn V → (∀𝑥𝐵 ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V → ∀𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V))
3736adantr 261 . . . . . . 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 5746 . . . . . 6 ((𝐵 ∈ On ∧ ∀𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V) → 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
4027, 38, 39syl2anc 391 . . . . 5 ((𝐹 Fn V ∧ 𝐵 ∈ On) → 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
41403adant2 923 . . . 4 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
42 unexg 4178 . . . . . 6 ((𝐴𝑉 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V) → (𝐴 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))) ∈ V)
4342ex 108 . . . . 5 (𝐴𝑉 → ( 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V → (𝐴 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))) ∈ V))
44433ad2ant2 926 . . . 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 5253 . 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 2072 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 97  w3a 885   = wceq 1243  wcel 1393  wral 2306  Vcvv 2557  cun 2915  wss 2917   ciun 3657  cmpt 3818  Oncon0 4100  dom cdm 4345  cres 4347  Fun wfun 4896   Fn wfn 4897  cfv 4902  reccrdg 5956
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 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-recs 5920  df-irdg 5957
This theorem is referenced by:  rdgival  5969  rdgon  5973
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