Theorem nfcoll 44218

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 44213	. 2 ⊢ (𝐹 Coll 𝐴) = ∪ 𝑦 ∈ 𝐴 Scott (𝐹 “ {𝑦})
2		nfcoll.2	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfcoll.1	. . . . 5 ⊢ Ⅎ𝑥𝐹
4		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑥{𝑦}
5	3, 4	nfima 6028	. . . 4 ⊢ Ⅎ𝑥(𝐹 “ {𝑦})
6	5	nfscott 44201	. . 3 ⊢ Ⅎ𝑥Scott (𝐹 “ {𝑦})
7	2, 6	nfiun 4983	. 2 ⊢ Ⅎ𝑥∪ 𝑦 ∈ 𝐴 Scott (𝐹 “ {𝑦})
8	1, 7	nfcxfr 2889	1 ⊢ Ⅎ𝑥(𝐹 Coll 𝐴)

Syntax hints:  Ⅎwnfc 2876  {csn 4585  ∪ ciun 4951   “ cima 5634  Scott cscott 44197   Coll ccoll 44212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-iun 4953  df-br 5103  df-opab 5165  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-scott 44198  df-coll 44213

This theorem is referenced by: (None)