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Theorem setrec2v 45165
Description: Version of setrec2 45164 with a disjoint variable 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 2979 . 2 𝑎𝐹
2 setrec2.b . 2 𝐵 = setrecs(𝐹)
3 setrec2.c . 2 (𝜑 → ∀𝑎(𝑎𝐶 → (𝐹𝑎) ⊆ 𝐶))
41, 2, 3setrec2 45164 1 (𝜑𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536   = wceq 1538  wss 3908  cfv 6334  setrecscsetrecs 45152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fv 6342  df-setrecs 45153
This theorem is referenced by:  setis  45166  elsetrecslem  45167  setrecsss  45169  setrecsres  45170  0setrec  45172  onsetrec  45176
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