Theorem nfcoll 43015

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 43010	. 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 6068	. . . 4 ⊢ Ⅎ𝑥(𝐹 “ {𝑦})
6	5	nfscott 42998	. . 3 ⊢ Ⅎ𝑥Scott (𝐹 “ {𝑦})
7	2, 6	nfiun 5028	. 2 ⊢ Ⅎ𝑥∪ 𝑦 ∈ 𝐴 Scott (𝐹 “ {𝑦})
8	1, 7	nfcxfr 2902	1 ⊢ Ⅎ𝑥(𝐹 Coll 𝐴)

Syntax hints:  Ⅎwnfc 2884  {csn 4629  ∪ ciun 4998   “ cima 5680  Scott cscott 42994   Coll ccoll 43009

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 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-iun 5000  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-scott 42995  df-coll 43010

This theorem is referenced by: (None)