Theorem dssmapfv2d 44445

Description: Value of the duality operator for self-mappings of subsets of a base set, 𝐵 when applied to function 𝐹. (Contributed by RP, 19-Apr-2021.)

Hypotheses

Ref	Expression
dssmapfvd.o	⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))
dssmapfvd.d	⊢ 𝐷 = (𝑂‘𝐵)
dssmapfvd.b	⊢ (𝜑 → 𝐵 ∈ 𝑉)
dssmapfv2d.f	⊢ (𝜑 → 𝐹 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
dssmapfv2d.g	⊢ 𝐺 = (𝐷‘𝐹)

Assertion

Ref	Expression
dssmapfv2d	⊢ (𝜑 → 𝐺 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑠)))))

Distinct variable groups:   𝐵,𝑏,𝑓,𝑠   𝑓,𝐹,𝑠   𝜑,𝑏,𝑓

Allowed substitution hints:   𝜑(𝑠)   𝐷(𝑓,𝑠,𝑏)   𝐹(𝑏)   𝐺(𝑓,𝑠,𝑏)   𝑂(𝑓,𝑠,𝑏)   𝑉(𝑓,𝑠,𝑏)

Proof of Theorem dssmapfv2d

Step	Hyp	Ref	Expression
1		dssmapfv2d.g	. 2 ⊢ 𝐺 = (𝐷‘𝐹)
2		dssmapfvd.o	. . . 4 ⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))
3		dssmapfvd.d	. . . 4 ⊢ 𝐷 = (𝑂‘𝐵)
4		dssmapfvd.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
5	2, 3, 4	dssmapfvd 44444	. . 3 ⊢ (𝜑 → 𝐷 = (𝑓 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))))
6		fveq1 6840	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝐵 ∖ 𝑠)) = (𝐹‘(𝐵 ∖ 𝑠)))
7	6	difeq2d 4067	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))) = (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑠))))
8	7	mpteq2dv 5180	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠)))) = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑠)))))
9	8	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠)))) = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑠)))))
10		dssmapfv2d.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
11		pwexg 5321	. . . 4 ⊢ (𝐵 ∈ 𝑉 → 𝒫 𝐵 ∈ V)
12		mptexg 7176	. . . 4 ⊢ (𝒫 𝐵 ∈ V → (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑠)))) ∈ V)
13	4, 11, 12	3syl 18	. . 3 ⊢ (𝜑 → (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑠)))) ∈ V)
14	5, 9, 10, 13	fvmptd 6956	. 2 ⊢ (𝜑 → (𝐷‘𝐹) = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑠)))))
15	1, 14	eqtrid 2784	1 ⊢ (𝜑 → 𝐺 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑠)))))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3430   ∖ cdif 3887  𝒫 cpw 4542   ↦ cmpt 5167  ‘cfv 6499  (class class class)co 7367   ↑_m cmap 8773

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 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7370

This theorem is referenced by:  dssmapfv3d  44446