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Theorem setrec2v 42968
Description: Version of setrec2 42967 with a dv condition instead of a non-freeness hypothesis. (Contributed by Emmett Weisz, 6-Mar-2021.)
Hypotheses
Ref Expression
setrec2.b 𝐵 = setrecs(𝐹)
setrec2.c (𝜑 → ∀𝑎(𝑎𝐶 → (𝐹𝑎) ⊆ 𝐶))
Assertion
Ref Expression
setrec2v (𝜑𝐵𝐶)
Distinct variable groups:   𝐹,𝑎   𝐶,𝑎
Allowed substitution hints:   𝜑(𝑎)   𝐵(𝑎)

Proof of Theorem setrec2v
StepHypRef Expression
1 nfcv 2913 . 2 𝑎𝐹
2 setrec2.b . 2 𝐵 = setrecs(𝐹)
3 setrec2.c . 2 (𝜑 → ∀𝑎(𝑎𝐶 → (𝐹𝑎) ⊆ 𝐶))
41, 2, 3setrec2 42967 1 (𝜑𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1629   = wceq 1631  wss 3723  cfv 6030  setrecscsetrecs 42955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fv 6038  df-setrecs 42956
This theorem is referenced by:  setis  42969  elsetrecslem  42970  setrecsss  42972  setrecsres  42973  0setrec  42975  onsetrec  42979
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