Theorem ntrclsss 44059

Description: If interior and closure functions are related then a subset relation of a pair of function values is equivalent to subset relation 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
ntrclsss	⊢ (𝜑 → ((𝐼‘𝑆) ⊆ (𝐼‘𝑇) ↔ (𝐾‘(𝐵 ∖ 𝑇)) ⊆ (𝐾‘(𝐵 ∖ 𝑆))))

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 ntrclsss

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.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)
5	1, 2, 3, 4	ntrclsfv 44055	. . 3 ⊢ (𝜑 → (𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))))
6		ntrclsfv.t	. . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝐵)
7	1, 2, 3, 6	ntrclsfv 44055	. . 3 ⊢ (𝜑 → (𝐼‘𝑇) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇))))
8	5, 7	sseq12d 3983	. 2 ⊢ (𝜑 → ((𝐼‘𝑆) ⊆ (𝐼‘𝑇) ↔ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) ⊆ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))))
9	1, 2, 3	ntrclskex 44050	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
10	9	ancli 548	. . 3 ⊢ (𝜑 → (𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)))
11		elmapi 8825	. . . . . . 7 ⊢ (𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
12	11	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
13	2, 3	ntrclsrcomplex 44031	. . . . . . 7 ⊢ (𝜑 → (𝐵 ∖ 𝑇) ∈ 𝒫 𝐵)
14	13	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝐵 ∖ 𝑇) ∈ 𝒫 𝐵)
15	12, 14	ffvelcdmd 7060	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝐾‘(𝐵 ∖ 𝑇)) ∈ 𝒫 𝐵)
16	15	elpwid 4575	. . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝐾‘(𝐵 ∖ 𝑇)) ⊆ 𝐵)
17	2, 3	ntrclsrcomplex 44031	. . . . . . 7 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)
18	17	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)
19	12, 18	ffvelcdmd 7060	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝐾‘(𝐵 ∖ 𝑆)) ∈ 𝒫 𝐵)
20	19	elpwid 4575	. . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝐾‘(𝐵 ∖ 𝑆)) ⊆ 𝐵)
21	16, 20	jca 511	. . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → ((𝐾‘(𝐵 ∖ 𝑇)) ⊆ 𝐵 ∧ (𝐾‘(𝐵 ∖ 𝑆)) ⊆ 𝐵))
22		sscon34b 4270	. . 3 ⊢ (((𝐾‘(𝐵 ∖ 𝑇)) ⊆ 𝐵 ∧ (𝐾‘(𝐵 ∖ 𝑆)) ⊆ 𝐵) → ((𝐾‘(𝐵 ∖ 𝑇)) ⊆ (𝐾‘(𝐵 ∖ 𝑆)) ↔ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) ⊆ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))))
23	10, 21, 22	3syl 18	. 2 ⊢ (𝜑 → ((𝐾‘(𝐵 ∖ 𝑇)) ⊆ (𝐾‘(𝐵 ∖ 𝑆)) ↔ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) ⊆ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))))
24	8, 23	bitr4d 282	1 ⊢ (𝜑 → ((𝐼‘𝑆) ⊆ (𝐼‘𝑇) ↔ (𝐾‘(𝐵 ∖ 𝑇)) ⊆ (𝐾‘(𝐵 ∖ 𝑆))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ∖ cdif 3914   ⊆ wss 3917  𝒫 cpw 4566   class class class wbr 5110   ↦ cmpt 5191  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ↑_m cmap 8802

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-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  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-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-map 8804

This theorem is referenced by: (None)