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Theorem rdgellim 35066
 Description: Elementhood in a recursive definition at a limit ordinal. (Contributed by ML, 30-Mar-2022.)
Assertion
Ref Expression
rdgellim (((𝐵 ∈ On ∧ Lim 𝐵) ∧ 𝐶𝐵) → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶) → 𝑋 ∈ (rec(𝐹, 𝐴)‘𝐵)))

Proof of Theorem rdgellim
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6659 . . . . . . 7 (𝑦 = 𝐶 → (rec(𝐹, 𝐴)‘𝑦) = (rec(𝐹, 𝐴)‘𝐶))
21eleq2d 2838 . . . . . 6 (𝑦 = 𝐶 → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝑦) ↔ 𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶)))
32rspcev 3542 . . . . 5 ((𝐶𝐵𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶)) → ∃𝑦𝐵 𝑋 ∈ (rec(𝐹, 𝐴)‘𝑦))
43ex 417 . . . 4 (𝐶𝐵 → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶) → ∃𝑦𝐵 𝑋 ∈ (rec(𝐹, 𝐴)‘𝑦)))
5 eliun 4888 . . . 4 (𝑋 𝑦𝐵 (rec(𝐹, 𝐴)‘𝑦) ↔ ∃𝑦𝐵 𝑋 ∈ (rec(𝐹, 𝐴)‘𝑦))
64, 5syl6ibr 255 . . 3 (𝐶𝐵 → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶) → 𝑋 𝑦𝐵 (rec(𝐹, 𝐴)‘𝑦)))
76adantl 486 . 2 (((𝐵 ∈ On ∧ Lim 𝐵) ∧ 𝐶𝐵) → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶) → 𝑋 𝑦𝐵 (rec(𝐹, 𝐴)‘𝑦)))
8 rdglim2a 8080 . . . 4 ((𝐵 ∈ On ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = 𝑦𝐵 (rec(𝐹, 𝐴)‘𝑦))
98eleq2d 2838 . . 3 ((𝐵 ∈ On ∧ Lim 𝐵) → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐵) ↔ 𝑋 𝑦𝐵 (rec(𝐹, 𝐴)‘𝑦)))
109adantr 485 . 2 (((𝐵 ∈ On ∧ Lim 𝐵) ∧ 𝐶𝐵) → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐵) ↔ 𝑋 𝑦𝐵 (rec(𝐹, 𝐴)‘𝑦)))
117, 10sylibrd 262 1 (((𝐵 ∈ On ∧ Lim 𝐵) ∧ 𝐶𝐵) → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶) → 𝑋 ∈ (rec(𝐹, 𝐴)‘𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1539   ∈ wcel 2112  ∃wrex 3072  ∪ ciun 4884  Oncon0 6170  Lim wlim 6171  ‘cfv 6336  reccrdg 8056 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-wrecs 7958  df-recs 8019  df-rdg 8057 This theorem is referenced by:  rdglimss  35067  exrecfnlem  35069
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