Theorem ntrclsfveq 44576

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 44573	. . 3 ⊢ (𝜑 → (𝐼‘𝑇) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇))))
6	5	eqeq2d 2763	. 2 ⊢ (𝜑 → ((𝐼‘𝑆) = (𝐼‘𝑇) ↔ (𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))))
7		ntrclsfv.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)
8	2, 3	ntrclsrcomplex 44549	. . 3 ⊢ (𝜑 → (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇))) ∈ 𝒫 𝐵)
9	1, 2, 3, 7, 8	ntrclsfveq1 44574	. 2 ⊢ (𝜑 → ((𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇))) ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇))))))
10	1, 2, 3	ntrclskex 44568	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
11		elmapi 8815	. . . . . . 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 44549	. . . . . 6 ⊢ (𝜑 → (𝐵 ∖ 𝑇) ∈ 𝒫 𝐵)
14	12, 13	ffvelcdmd 7051	. . . . 5 ⊢ (𝜑 → (𝐾‘(𝐵 ∖ 𝑇)) ∈ 𝒫 𝐵)
15	14	elpwid 4554	. . . 4 ⊢ (𝜑 → (𝐾‘(𝐵 ∖ 𝑇)) ⊆ 𝐵)
16		dfss4 4212	. . . 4 ⊢ ((𝐾‘(𝐵 ∖ 𝑇)) ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))) = (𝐾‘(𝐵 ∖ 𝑇)))
17	15, 16	sylib 220	. . 3 ⊢ (𝜑 → (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))) = (𝐾‘(𝐵 ∖ 𝑇)))
18	17	eqeq2d 2763	. 2 ⊢ (𝜑 → ((𝐾‘(𝐵 ∖ 𝑆)) = (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))) ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐾‘(𝐵 ∖ 𝑇))))
19	6, 9, 18	3bitrd 307	1 ⊢ (𝜑 → ((𝐼‘𝑆) = (𝐼‘𝑇) ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐾‘(𝐵 ∖ 𝑇))))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1550   ∈ wcel 2132  Vcvv 3444   ∖ cdif 3892   ⊆ wss 3895  𝒫 cpw 4545   class class class wbr 5090   ↦ cmpt 5171  ⟶wf 6502  ‘cfv 6506  (class class class)co 7381   ↑_m cmap 8792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-id 5531  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-ov 7384  df-oprab 7385  df-mpo 7386  df-1st 7955  df-2nd 7956  df-map 8794

This theorem is referenced by: (None)