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Theorem nfcoll 42628
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 42623 . 2 (𝐹 Coll 𝐴) = 𝑦𝐴 Scott (𝐹 “ {𝑦})
2 nfcoll.2 . . 3 𝑥𝐴
3 nfcoll.1 . . . . 5 𝑥𝐹
4 nfcv 2904 . . . . 5 𝑥{𝑦}
53, 4nfima 6025 . . . 4 𝑥(𝐹 “ {𝑦})
65nfscott 42611 . . 3 𝑥Scott (𝐹 “ {𝑦})
72, 6nfiun 4988 . 2 𝑥 𝑦𝐴 Scott (𝐹 “ {𝑦})
81, 7nfcxfr 2902 1 𝑥(𝐹 Coll 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2884  {csn 4590   ciun 4958  cima 5640  Scott cscott 42607   Coll ccoll 42622
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
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-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-iun 4960  df-br 5110  df-opab 5172  df-xp 5643  df-cnv 5645  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-scott 42608  df-coll 42623
This theorem is referenced by: (None)
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