Theorem nfcoll 44280

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 44275	. 2 ⊢ (𝐹 Coll 𝐴) = ∪ 𝑦 ∈ 𝐴 Scott (𝐹 “ {𝑦})
2		nfcoll.2	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfcoll.1	. . . . 5 ⊢ Ⅎ𝑥𝐹
4		nfcv 2904	. . . . 5 ⊢ Ⅎ𝑥{𝑦}
5	3, 4	nfima 6085	. . . 4 ⊢ Ⅎ𝑥(𝐹 “ {𝑦})
6	5	nfscott 44263	. . 3 ⊢ Ⅎ𝑥Scott (𝐹 “ {𝑦})
7	2, 6	nfiun 5022	. 2 ⊢ Ⅎ𝑥∪ 𝑦 ∈ 𝐴 Scott (𝐹 “ {𝑦})
8	1, 7	nfcxfr 2902	1 ⊢ Ⅎ𝑥(𝐹 Coll 𝐴)

Syntax hints:  Ⅎwnfc 2889  {csn 4625  ∪ ciun 4990   “ cima 5687  Scott cscott 44259   Coll ccoll 44274

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-iun 4992  df-br 5143  df-opab 5205  df-xp 5690  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-scott 44260  df-coll 44275

This theorem is referenced by: (None)