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Theorem setrec2v 47227
Description: Version of setrec2 47226 with a disjoint variable condition instead of a nonfreeness 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 2904 . 2 𝑎𝐹
2 setrec2.b . 2 𝐵 = setrecs(𝐹)
3 setrec2.c . 2 (𝜑 → ∀𝑎(𝑎𝐶 → (𝐹𝑎) ⊆ 𝐶))
41, 2, 3setrec2 47226 1 (𝜑𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1540   = wceq 1542  wss 3911  cfv 6497  setrecscsetrecs 47214
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fv 6505  df-setrecs 47215
This theorem is referenced by:  setis  47229  elsetrecslem  47230  setrecsss  47232  setrecsres  47233  0setrec  47235  onsetrec  47239
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