Theorem nfcoll 44295

Description: Bound-variable hypothesis builder for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)

Hypotheses

Ref	Expression
nfcoll.1	⊢ Ⅎ𝑥𝐹
nfcoll.2	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfcoll	⊢ Ⅎ𝑥(𝐹 Coll 𝐴)

Proof of Theorem nfcoll

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-coll 44290	. 2 ⊢ (𝐹 Coll 𝐴) = ∪ 𝑦 ∈ 𝐴 Scott (𝐹 “ {𝑦})
2		nfcoll.2	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfcoll.1	. . . . 5 ⊢ Ⅎ𝑥𝐹
4		nfcv 2894	. . . . 5 ⊢ Ⅎ𝑥{𝑦}
5	3, 4	nfima 6017	. . . 4 ⊢ Ⅎ𝑥(𝐹 “ {𝑦})
6	5	nfscott 44278	. . 3 ⊢ Ⅎ𝑥Scott (𝐹 “ {𝑦})
7	2, 6	nfiun 4973	. 2 ⊢ Ⅎ𝑥∪ 𝑦 ∈ 𝐴 Scott (𝐹 “ {𝑦})
8	1, 7	nfcxfr 2892	1 ⊢ Ⅎ𝑥(𝐹 Coll 𝐴)

Syntax hints:  Ⅎwnfc 2879  {csn 4576  ∪ ciun 4941   “ cima 5619  Scott cscott 44274   Coll ccoll 44289

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-iun 4943  df-br 5092  df-opab 5154  df-xp 5622  df-cnv 5624  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-scott 44275  df-coll 44290

This theorem is referenced by: (None)