Theorem neicvgrcomplex 44109

Description: The relative complement of the class 𝑆 exists as a subset of the base set. (Contributed by RP, 26-Jun-2021.)

Hypotheses

Ref	Expression
neicvgbex.d	⊢ 𝐷 = (𝑃‘𝐵)
neicvgbex.h	⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))
neicvgbex.r	⊢ (𝜑 → 𝑁𝐻𝑀)

Assertion

Ref	Expression
neicvgrcomplex	⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)

Proof of Theorem neicvgrcomplex

Step	Hyp	Ref	Expression
1		neicvgbex.d	. . 3 ⊢ 𝐷 = (𝑃‘𝐵)
2		neicvgbex.h	. . 3 ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))
3		neicvgbex.r	. . 3 ⊢ (𝜑 → 𝑁𝐻𝑀)
4	1, 2, 3	neicvgbex 44108	. 2 ⊢ (𝜑 → 𝐵 ∈ V)
5		difssd 4103	. 2 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ⊆ 𝐵)
6	4, 5	sselpwd 5286	1 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ∖ cdif 3914  𝒫 cpw 4566   class class class wbr 5110   ∘ ccom 5645  ‘cfv 6514

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-iota 6467  df-fv 6522

This theorem is referenced by:  neicvgel2  44116