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Theorem nfcoll 40884
 Description: Bound-variable hypothesis builder for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
Hypotheses
Ref Expression
nfcoll.1 𝑥𝐹
nfcoll.2 𝑥𝐴
Assertion
Ref Expression
nfcoll 𝑥(𝐹 Coll 𝐴)

Proof of Theorem nfcoll
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-coll 40879 . 2 (𝐹 Coll 𝐴) = 𝑦𝐴 Scott (𝐹 “ {𝑦})
2 nfcoll.2 . . 3 𝑥𝐴
3 nfcoll.1 . . . . 5 𝑥𝐹
4 nfcv 2982 . . . . 5 𝑥{𝑦}
53, 4nfima 5924 . . . 4 𝑥(𝐹 “ {𝑦})
65nfscott 40867 . . 3 𝑥Scott (𝐹 “ {𝑦})
72, 6nfiun 4935 . 2 𝑥 𝑦𝐴 Scott (𝐹 “ {𝑦})
81, 7nfcxfr 2980 1 𝑥(𝐹 Coll 𝐴)
 Colors of variables: wff setvar class Syntax hints:  Ⅎwnfc 2962  {csn 4550  ∪ ciun 4905   “ cima 5545  Scott cscott 40863   Coll ccoll 40878 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796 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 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-iun 4907  df-br 5053  df-opab 5115  df-xp 5548  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-scott 40864  df-coll 40879 This theorem is referenced by: (None)
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