Theorem ntrclsfveq 38360

Description: If interior and closure functions are related then equality of a pair of function values is equivalent to equality of a pair of the other function's values. (Contributed by RP, 27-Jun-2021.)

Hypotheses

Ref	Expression
ntrcls.o	⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
ntrcls.d	⊢ 𝐷 = (𝑂‘𝐵)
ntrcls.r	⊢ (𝜑 → 𝐼𝐷𝐾)
ntrclsfv.s	⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)
ntrclsfv.t	⊢ (𝜑 → 𝑇 ∈ 𝒫 𝐵)

Assertion

Ref	Expression
ntrclsfveq	⊢ (𝜑 → ((𝐼‘𝑆) = (𝐼‘𝑇) ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐾‘(𝐵 ∖ 𝑇))))

Distinct variable groups:   𝐵,𝑖,𝑗,𝑘   𝑗,𝐾,𝑘   𝑆,𝑗   𝑇,𝑗   𝜑,𝑖,𝑗,𝑘

Allowed substitution hints:   𝐷(𝑖,𝑗,𝑘)   𝑆(𝑖,𝑘)   𝑇(𝑖,𝑘)   𝐼(𝑖,𝑗,𝑘)   𝐾(𝑖)   𝑂(𝑖,𝑗,𝑘)

Proof of Theorem ntrclsfveq

Step	Hyp	Ref	Expression
1		ntrcls.o	. . . 4 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
2		ntrcls.d	. . . 4 ⊢ 𝐷 = (𝑂‘𝐵)
3		ntrcls.r	. . . 4 ⊢ (𝜑 → 𝐼𝐷𝐾)
4		ntrclsfv.t	. . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝐵)
5	1, 2, 3, 4	ntrclsfv 38357	. . 3 ⊢ (𝜑 → (𝐼‘𝑇) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇))))
6	5	eqeq2d 2632	. 2 ⊢ (𝜑 → ((𝐼‘𝑆) = (𝐼‘𝑇) ↔ (𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))))
7		ntrclsfv.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)
8	2, 3	ntrclsrcomplex 38333	. . 3 ⊢ (𝜑 → (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇))) ∈ 𝒫 𝐵)
9	1, 2, 3, 7, 8	ntrclsfveq1 38358	. 2 ⊢ (𝜑 → ((𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇))) ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇))))))
10	1, 2, 3	ntrclskex 38352	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵))
11		elmapi 7879	. . . . . . 7 ⊢ (𝐾 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
12	10, 11	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
13	2, 3	ntrclsrcomplex 38333	. . . . . 6 ⊢ (𝜑 → (𝐵 ∖ 𝑇) ∈ 𝒫 𝐵)
14	12, 13	ffvelrnd 6360	. . . . 5 ⊢ (𝜑 → (𝐾‘(𝐵 ∖ 𝑇)) ∈ 𝒫 𝐵)
15	14	elpwid 4170	. . . 4 ⊢ (𝜑 → (𝐾‘(𝐵 ∖ 𝑇)) ⊆ 𝐵)
16		dfss4 3858	. . . 4 ⊢ ((𝐾‘(𝐵 ∖ 𝑇)) ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))) = (𝐾‘(𝐵 ∖ 𝑇)))
17	15, 16	sylib 208	. . 3 ⊢ (𝜑 → (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))) = (𝐾‘(𝐵 ∖ 𝑇)))
18	17	eqeq2d 2632	. 2 ⊢ (𝜑 → ((𝐾‘(𝐵 ∖ 𝑆)) = (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))) ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐾‘(𝐵 ∖ 𝑇))))
19	6, 9, 18	3bitrd 294	1 ⊢ (𝜑 → ((𝐼‘𝑆) = (𝐼‘𝑇) ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐾‘(𝐵 ∖ 𝑇))))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  𝒫 cpw 4158   class class class wbr 4653   ↦ cmpt 4729  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859

This theorem is referenced by: (None)