Theorem nfcoll 44966

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 44961	. 2 ⊢ (𝐹 Coll 𝐴) = ∪ 𝑦 ∈ 𝐴 Scott (𝐹 “ {𝑦})
2		nfcoll.2	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfcoll.1	. . . . 5 ⊢ Ⅎ𝑥𝐹
4		nfcv 2925	. . . . 5 ⊢ Ⅎ𝑥{𝑦}
5	3, 4	nfima 6070	. . . 4 ⊢ Ⅎ𝑥(𝐹 “ {𝑦})
6	5	nfscott 9861	. . 3 ⊢ Ⅎ𝑥Scott (𝐹 “ {𝑦})
7	2, 6	nfiun 4988	. 2 ⊢ Ⅎ𝑥∪ 𝑦 ∈ 𝐴 Scott (𝐹 “ {𝑦})
8	1, 7	nfcxfr 2923	1 ⊢ Ⅎ𝑥(𝐹 Coll 𝐴)

Syntax hints:  Ⅎwnfc 2910  {csn 4589  ∪ ciun 4956   “ cima 5664  Scott cscott 9853   Coll ccoll 44960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-iun 4958  df-br 5110  df-opab 5174  df-xp 5667  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-scott 9854  df-coll 44961

This theorem is referenced by: (None)