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Theorem rdglimss 37292
Description: A recursive definition at a limit ordinal is a superset of itself at any smaller ordinal. (Contributed by ML, 30-Mar-2022.)
Assertion
Ref Expression
rdglimss (((𝐵 ∈ On ∧ Lim 𝐵) ∧ 𝐶𝐵) → (rec(𝐹, 𝐴)‘𝐶) ⊆ (rec(𝐹, 𝐴)‘𝐵))

Proof of Theorem rdglimss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 rdgellim 37291 . 2 (((𝐵 ∈ On ∧ Lim 𝐵) ∧ 𝐶𝐵) → (𝑥 ∈ (rec(𝐹, 𝐴)‘𝐶) → 𝑥 ∈ (rec(𝐹, 𝐴)‘𝐵)))
21ssrdv 4008 1 (((𝐵 ∈ On ∧ Lim 𝐵) ∧ 𝐶𝐵) → (rec(𝐹, 𝐴)‘𝐶) ⊆ (rec(𝐹, 𝐴)‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2103  wss 3970  Oncon0 6394  Lim wlim 6395  cfv 6572  reccrdg 8461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pr 5450  ax-un 7766
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-pred 6331  df-ord 6397  df-on 6398  df-lim 6399  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-ov 7448  df-2nd 8027  df-frecs 8318  df-wrecs 8349  df-recs 8423  df-rdg 8462
This theorem is referenced by:  exrecfnlem  37294
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