Theorem ntrclsss 44478

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 44474	. . 3 ⊢ (𝜑 → (𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))))
6		ntrclsfv.t	. . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝐵)
7	1, 2, 3, 6	ntrclsfv 44474	. . 3 ⊢ (𝜑 → (𝐼‘𝑇) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇))))
8	5, 7	sseq12d 3950	. 2 ⊢ (𝜑 → ((𝐼‘𝑆) ⊆ (𝐼‘𝑇) ↔ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) ⊆ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))))
9	1, 2, 3	ntrclskex 44469	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
10	9	ancli 548	. . 3 ⊢ (𝜑 → (𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)))
11		elmapi 8785	. . . . . . 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 44450	. . . . . . 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 7026	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝐾‘(𝐵 ∖ 𝑇)) ∈ 𝒫 𝐵)
16	15	elpwid 4540	. . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝐾‘(𝐵 ∖ 𝑇)) ⊆ 𝐵)
17	2, 3	ntrclsrcomplex 44450	. . . . . . 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 7026	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝐾‘(𝐵 ∖ 𝑆)) ∈ 𝒫 𝐵)
20	19	elpwid 4540	. . . 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 4234	. . 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 1542   ∈ wcel 2114  Vcvv 3427   ∖ cdif 3882   ⊆ wss 3885  𝒫 cpw 4531   class class class wbr 5074   ↦ cmpt 5155  ⟶wf 6483  ‘cfv 6487  (class class class)co 7356   ↑_m cmap 8762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-map 8764

This theorem is referenced by: (None)