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Theorem rdglim2a 7393
Description: The value of the recursive definition generator at a limit ordinal, in terms of indexed union of all smaller values. (Contributed by NM, 28-Jun-1998.)
Assertion
Ref Expression
rdglim2a ((𝐵𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = 𝑥𝐵 (rec(𝐹, 𝐴)‘𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem rdglim2a
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 rdglim2 7392 . 2 ((𝐵𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
2 fvex 6097 . . 3 (rec(𝐹, 𝐴)‘𝑥) ∈ V
32dfiun2 4484 . 2 𝑥𝐵 (rec(𝐹, 𝐴)‘𝑥) = {𝑦 ∣ ∃𝑥𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)}
41, 3syl6eqr 2661 1 ((𝐵𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = 𝑥𝐵 (rec(𝐹, 𝐴)‘𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  {cab 2595  wrex 2896   cuni 4366   ciun 4449  Lim wlim 5626  cfv 5789  reccrdg 7369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-wrecs 7271  df-recs 7332  df-rdg 7370
This theorem is referenced by:  oalim  7476  omlim  7477  oelim  7478  alephlim  8750
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