Theorem pwrssmgc 33440

Description: Given a function 𝐹, exhibit a Galois connection between subsets of its domain and subsets of its range. (Contributed by Thierry Arnoux, 26-Apr-2024.)

Hypotheses

Ref	Expression
pwrssmgc.1	⊢ 𝐺 = (𝑛 ∈ 𝒫 𝑌 ↦ (◡𝐹 “ 𝑛))
pwrssmgc.2	⊢ 𝐻 = (𝑚 ∈ 𝒫 𝑋 ↦ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑚})
pwrssmgc.3	⊢ 𝑉 = (toInc‘𝒫 𝑌)
pwrssmgc.4	⊢ 𝑊 = (toInc‘𝒫 𝑋)
pwrssmgc.5	⊢ (𝜑 → 𝑋 ∈ 𝐴)
pwrssmgc.6	⊢ (𝜑 → 𝑌 ∈ 𝐵)
pwrssmgc.7	⊢ (𝜑 → 𝐹:𝑋⟶𝑌)

Assertion

Ref	Expression
pwrssmgc	⊢ (𝜑 → 𝐺(𝑉MGalConn𝑊)𝐻)

Distinct variable groups:   𝑚,𝐹,𝑦   𝑛,𝐹   𝑚,𝑉,𝑦   𝑛,𝑉   𝑚,𝑊,𝑦   𝑛,𝑊   𝑚,𝑋   𝑛,𝑋   𝑚,𝑌,𝑦   𝑛,𝑌   𝜑,𝑦,𝑚   𝜑,𝑛

Allowed substitution hints:   𝐴(𝑦, 𝑚, 𝑛)   𝐵(𝑦, 𝑚, 𝑛)   𝐺(𝑦, 𝑚, 𝑛)   𝐻(𝑦, 𝑚, 𝑛)   𝑋(𝑦)

Proof of Theorem pwrssmgc

Dummy variables 𝑖 𝑗 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwrssmgc.5	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
2	1	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝒫 𝑌) → 𝑋 ∈ 𝐴)
3		cnvimass 6078	. . . . . . . 8 ⊢ (◡𝐹 “ 𝑛) ⊆ dom 𝐹
4		pwrssmgc.7	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)
5	3, 4	fssdm 6722	. . . . . . 7 ⊢ (𝜑 → (◡𝐹 “ 𝑛) ⊆ 𝑋)
6	5	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝒫 𝑌) → (◡𝐹 “ 𝑛) ⊆ 𝑋)
7	2, 6	sselpwd 5293	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝒫 𝑌) → (◡𝐹 “ 𝑛) ∈ 𝒫 𝑋)
8		pwrssmgc.1	. . . . 5 ⊢ 𝐺 = (𝑛 ∈ 𝒫 𝑌 ↦ (◡𝐹 “ 𝑛))
9	7, 8	fmptd 7107	. . . 4 ⊢ (𝜑 → 𝐺:𝒫 𝑌⟶𝒫 𝑋)
10		pwrssmgc.6	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
11		pwexg 5343	. . . . . 6 ⊢ (𝑌 ∈ 𝐵 → 𝒫 𝑌 ∈ V)
12		pwrssmgc.3	. . . . . . 7 ⊢ 𝑉 = (toInc‘𝒫 𝑌)
13	12	ipobas 18619	. . . . . 6 ⊢ (𝒫 𝑌 ∈ V → 𝒫 𝑌 = (Base‘𝑉))
14	10, 11, 13	3syl 19	. . . . 5 ⊢ (𝜑 → 𝒫 𝑌 = (Base‘𝑉))
15		pwexg 5343	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → 𝒫 𝑋 ∈ V)
16		pwrssmgc.4	. . . . . . 7 ⊢ 𝑊 = (toInc‘𝒫 𝑋)
17	16	ipobas 18619	. . . . . 6 ⊢ (𝒫 𝑋 ∈ V → 𝒫 𝑋 = (Base‘𝑊))
18	1, 15, 17	3syl 19	. . . . 5 ⊢ (𝜑 → 𝒫 𝑋 = (Base‘𝑊))
19	14, 18	feq23d 6697	. . . 4 ⊢ (𝜑 → (𝐺:𝒫 𝑌⟶𝒫 𝑋 ↔ 𝐺:(Base‘𝑉)⟶(Base‘𝑊)))
20	9, 19	mpbid 235	. . 3 ⊢ (𝜑 → 𝐺:(Base‘𝑉)⟶(Base‘𝑊))
21	10	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝒫 𝑋) → 𝑌 ∈ 𝐵)
22		ssrab2 4028	. . . . . . 7 ⊢ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑚} ⊆ 𝑌
23	22	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝒫 𝑋) → {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑚} ⊆ 𝑌)
24	21, 23	sselpwd 5293	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝒫 𝑋) → {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑚} ∈ 𝒫 𝑌)
25		pwrssmgc.2	. . . . 5 ⊢ 𝐻 = (𝑚 ∈ 𝒫 𝑋 ↦ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑚})
26	24, 25	fmptd 7107	. . . 4 ⊢ (𝜑 → 𝐻:𝒫 𝑋⟶𝒫 𝑌)
27	18, 14	feq23d 6697	. . . 4 ⊢ (𝜑 → (𝐻:𝒫 𝑋⟶𝒫 𝑌 ↔ 𝐻:(Base‘𝑊)⟶(Base‘𝑉)))
28	26, 27	mpbid 235	. . 3 ⊢ (𝜑 → 𝐻:(Base‘𝑊)⟶(Base‘𝑉))
29	20, 28	jca 521	. 2 ⊢ (𝜑 → (𝐺:(Base‘𝑉)⟶(Base‘𝑊) ∧ 𝐻:(Base‘𝑊)⟶(Base‘𝑉)))
30		sneq 4594	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑗 → {𝑦} = {𝑗})
31	30	imaeq2d 6056	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑗 → (◡𝐹 “ {𝑦}) = (◡𝐹 “ {𝑗}))
32	31	sseq1d 3962	. . . . . . . . . 10 ⊢ (𝑦 = 𝑗 → ((◡𝐹 “ {𝑦}) ⊆ 𝑣 ↔ (◡𝐹 “ {𝑗}) ⊆ 𝑣))
33		simplr 781	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑢 ∈ (Base‘𝑉))
34	14	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝒫 𝑌 = (Base‘𝑉))
35	33, 34	eleqtrrd 2863	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑢 ∈ 𝒫 𝑌)
36	35	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (◡𝐹 “ 𝑢) ⊆ 𝑣) → 𝑢 ∈ 𝒫 𝑌)
37	36	elpwid 4566	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (◡𝐹 “ 𝑢) ⊆ 𝑣) → 𝑢 ⊆ 𝑌)
38	37	sselda 3931	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (◡𝐹 “ 𝑢) ⊆ 𝑣) ∧ 𝑗 ∈ 𝑢) → 𝑗 ∈ 𝑌)
39	4	ffund 6707	. . . . . . . . . . . . 13 ⊢ (𝜑 → Fun 𝐹)
40	39	ad4antr 745	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (◡𝐹 “ 𝑢) ⊆ 𝑣) ∧ 𝑗 ∈ 𝑢) → Fun 𝐹)
41		snssi 4746	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ 𝑢 → {𝑗} ⊆ 𝑢)
42	41	adantl 487	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (◡𝐹 “ 𝑢) ⊆ 𝑣) ∧ 𝑗 ∈ 𝑢) → {𝑗} ⊆ 𝑢)
43		sspreima 7060	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ {𝑗} ⊆ 𝑢) → (◡𝐹 “ {𝑗}) ⊆ (◡𝐹 “ 𝑢))
44	40, 42, 43	syl2anc 596	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (◡𝐹 “ 𝑢) ⊆ 𝑣) ∧ 𝑗 ∈ 𝑢) → (◡𝐹 “ {𝑗}) ⊆ (◡𝐹 “ 𝑢))
45		simplr 781	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (◡𝐹 “ 𝑢) ⊆ 𝑣) ∧ 𝑗 ∈ 𝑢) → (◡𝐹 “ 𝑢) ⊆ 𝑣)
46	44, 45	sstrd 3941	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (◡𝐹 “ 𝑢) ⊆ 𝑣) ∧ 𝑗 ∈ 𝑢) → (◡𝐹 “ {𝑗}) ⊆ 𝑣)
47	32, 38, 46	elrabd 3647	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (◡𝐹 “ 𝑢) ⊆ 𝑣) ∧ 𝑗 ∈ 𝑢) → 𝑗 ∈ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣})
48	47	ex 418	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (◡𝐹 “ 𝑢) ⊆ 𝑣) → (𝑗 ∈ 𝑢 → 𝑗 ∈ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣}))
49	48	ssrdv 3937	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (◡𝐹 “ 𝑢) ⊆ 𝑣) → 𝑢 ⊆ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣})
50		simplr 781	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (◡𝐹 “ 𝑢)) → 𝑢 ⊆ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣})
51	4	ffnd 6703	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 Fn 𝑋)
52	51	ad4antr 745	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (◡𝐹 “ 𝑢)) → 𝐹 Fn 𝑋)
53		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (◡𝐹 “ 𝑢)) → 𝑖 ∈ (◡𝐹 “ 𝑢))
54		elpreima 7050	. . . . . . . . . . . . . . 15 ⊢ (𝐹 Fn 𝑋 → (𝑖 ∈ (◡𝐹 “ 𝑢) ↔ (𝑖 ∈ 𝑋 ∧ (𝐹‘𝑖) ∈ 𝑢)))
55	54	biimpa 482	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑖 ∈ (◡𝐹 “ 𝑢)) → (𝑖 ∈ 𝑋 ∧ (𝐹‘𝑖) ∈ 𝑢))
56	52, 53, 55	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (◡𝐹 “ 𝑢)) → (𝑖 ∈ 𝑋 ∧ (𝐹‘𝑖) ∈ 𝑢))
57	56	simprd 501	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (◡𝐹 “ 𝑢)) → (𝐹‘𝑖) ∈ 𝑢)
58	50, 57	sseldd 3932	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (◡𝐹 “ 𝑢)) → (𝐹‘𝑖) ∈ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣})
59		sneq 4594	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝐹‘𝑖) → {𝑦} = {(𝐹‘𝑖)})
60	59	imaeq2d 6056	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐹‘𝑖) → (◡𝐹 “ {𝑦}) = (◡𝐹 “ {(𝐹‘𝑖)}))
61	60	sseq1d 3962	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐹‘𝑖) → ((◡𝐹 “ {𝑦}) ⊆ 𝑣 ↔ (◡𝐹 “ {(𝐹‘𝑖)}) ⊆ 𝑣))
62	61	elrab 3645	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑖) ∈ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣} ↔ ((𝐹‘𝑖) ∈ 𝑌 ∧ (◡𝐹 “ {(𝐹‘𝑖)}) ⊆ 𝑣))
63	62	simprbi 503	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑖) ∈ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣} → (◡𝐹 “ {(𝐹‘𝑖)}) ⊆ 𝑣)
64	58, 63	syl 18	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (◡𝐹 “ 𝑢)) → (◡𝐹 “ {(𝐹‘𝑖)}) ⊆ 𝑣)
65	56	simpld 500	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (◡𝐹 “ 𝑢)) → 𝑖 ∈ 𝑋)
66		eqidd 2761	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (◡𝐹 “ 𝑢)) → (𝐹‘𝑖) = (𝐹‘𝑖))
67		fniniseg 7052	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝑋 → (𝑖 ∈ (◡𝐹 “ {(𝐹‘𝑖)}) ↔ (𝑖 ∈ 𝑋 ∧ (𝐹‘𝑖) = (𝐹‘𝑖))))
68	67	biimpar 483	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝑋 ∧ (𝑖 ∈ 𝑋 ∧ (𝐹‘𝑖) = (𝐹‘𝑖))) → 𝑖 ∈ (◡𝐹 “ {(𝐹‘𝑖)}))
69	52, 65, 66, 68	syl12anc 850	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (◡𝐹 “ 𝑢)) → 𝑖 ∈ (◡𝐹 “ {(𝐹‘𝑖)}))
70	64, 69	sseldd 3932	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (◡𝐹 “ 𝑢)) → 𝑖 ∈ 𝑣)
71	70	ex 418	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣}) → (𝑖 ∈ (◡𝐹 “ 𝑢) → 𝑖 ∈ 𝑣))
72	71	ssrdv 3937	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣}) → (◡𝐹 “ 𝑢) ⊆ 𝑣)
73	49, 72	impbida 813	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → ((◡𝐹 “ 𝑢) ⊆ 𝑣 ↔ 𝑢 ⊆ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣}))
74		simpr 490	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑛 = 𝑢) → 𝑛 = 𝑢)
75	74	imaeq2d 6056	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑛 = 𝑢) → (◡𝐹 “ 𝑛) = (◡𝐹 “ 𝑢))
76	4, 1	fexd 7226	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ V)
77		cnvexg 7921	. . . . . . . . . 10 ⊢ (𝐹 ∈ V → ◡𝐹 ∈ V)
78		imaexg 7910	. . . . . . . . . 10 ⊢ (◡𝐹 ∈ V → (◡𝐹 “ 𝑢) ∈ V)
79	76, 77, 78	3syl 19	. . . . . . . . 9 ⊢ (𝜑 → (◡𝐹 “ 𝑢) ∈ V)
80	79	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → (◡𝐹 “ 𝑢) ∈ V)
81	8, 75, 35, 80	fvmptd2 6995	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝐺‘𝑢) = (◡𝐹 “ 𝑢))
82	81	sseq1d 3962	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → ((𝐺‘𝑢) ⊆ 𝑣 ↔ (◡𝐹 “ 𝑢) ⊆ 𝑣))
83		simpr 490	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑚 = 𝑣) → 𝑚 = 𝑣)
84	83	sseq2d 3963	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑚 = 𝑣) → ((◡𝐹 “ {𝑦}) ⊆ 𝑚 ↔ (◡𝐹 “ {𝑦}) ⊆ 𝑣))
85	84	rabbidv 3419	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑚 = 𝑣) → {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑚} = {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣})
86		simpr 490	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑣 ∈ (Base‘𝑊))
87	1, 15	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝒫 𝑋 ∈ V)
88	87	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝒫 𝑋 ∈ V)
89	88, 17	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝒫 𝑋 = (Base‘𝑊))
90	86, 89	eleqtrrd 2863	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑣 ∈ 𝒫 𝑋)
91	10	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑌 ∈ 𝐵)
92		ssrab2 4028	. . . . . . . . . 10 ⊢ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣} ⊆ 𝑌
93	92	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣} ⊆ 𝑌)
94	91, 93	sselpwd 5293	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣} ∈ 𝒫 𝑌)
95	25, 85, 90, 94	fvmptd2 6995	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝐻‘𝑣) = {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣})
96	95	sseq2d 3963	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝑢 ⊆ (𝐻‘𝑣) ↔ 𝑢 ⊆ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣}))
97	73, 82, 96	3bitr4d 314	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → ((𝐺‘𝑢) ⊆ 𝑣 ↔ 𝑢 ⊆ (𝐻‘𝑣)))
98	9	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝐺:𝒫 𝑌⟶𝒫 𝑋)
99	98, 35	ffvelcdmd 7078	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝐺‘𝑢) ∈ 𝒫 𝑋)
100		eqid 2760	. . . . . . 7 ⊢ (le‘𝑊) = (le‘𝑊)
101	16, 100	ipole 18622	. . . . . 6 ⊢ ((𝒫 𝑋 ∈ V ∧ (𝐺‘𝑢) ∈ 𝒫 𝑋 ∧ 𝑣 ∈ 𝒫 𝑋) → ((𝐺‘𝑢)(le‘𝑊)𝑣 ↔ (𝐺‘𝑢) ⊆ 𝑣))
102	88, 99, 90, 101	syl3anc 1398	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → ((𝐺‘𝑢)(le‘𝑊)𝑣 ↔ (𝐺‘𝑢) ⊆ 𝑣))
103	10, 11	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝒫 𝑌 ∈ V)
104	103	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝒫 𝑌 ∈ V)
105	26	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝐻:𝒫 𝑋⟶𝒫 𝑌)
106	105, 90	ffvelcdmd 7078	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝐻‘𝑣) ∈ 𝒫 𝑌)
107		eqid 2760	. . . . . . 7 ⊢ (le‘𝑉) = (le‘𝑉)
108	12, 107	ipole 18622	. . . . . 6 ⊢ ((𝒫 𝑌 ∈ V ∧ 𝑢 ∈ 𝒫 𝑌 ∧ (𝐻‘𝑣) ∈ 𝒫 𝑌) → (𝑢(le‘𝑉)(𝐻‘𝑣) ↔ 𝑢 ⊆ (𝐻‘𝑣)))
109	104, 35, 106, 108	syl3anc 1398	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝑢(le‘𝑉)(𝐻‘𝑣) ↔ 𝑢 ⊆ (𝐻‘𝑣)))
110	97, 102, 109	3bitr4d 314	. . . 4 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → ((𝐺‘𝑢)(le‘𝑊)𝑣 ↔ 𝑢(le‘𝑉)(𝐻‘𝑣)))
111	110	anasss 472	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑉) ∧ 𝑣 ∈ (Base‘𝑊))) → ((𝐺‘𝑢)(le‘𝑊)𝑣 ↔ 𝑢(le‘𝑉)(𝐻‘𝑣)))
112	111	ralrimivva 3205	. 2 ⊢ (𝜑 → ∀𝑢 ∈ (Base‘𝑉)∀𝑣 ∈ (Base‘𝑊)((𝐺‘𝑢)(le‘𝑊)𝑣 ↔ 𝑢(le‘𝑉)(𝐻‘𝑣)))
113		eqid 2760	. . 3 ⊢ (Base‘𝑉) = (Base‘𝑉)
114		eqid 2760	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
115		eqid 2760	. . 3 ⊢ (𝑉MGalConn𝑊) = (𝑉MGalConn𝑊)
116	12	ipopos 18624	. . . 4 ⊢ 𝑉 ∈ Poset
117		posprs 18404	. . . 4 ⊢ (𝑉 ∈ Poset → 𝑉 ∈ Proset )
118	116, 117	mp1i 14	. . 3 ⊢ (𝜑 → 𝑉 ∈ Proset )
119	16	ipopos 18624	. . . 4 ⊢ 𝑊 ∈ Poset
120		posprs 18404	. . . 4 ⊢ (𝑊 ∈ Poset → 𝑊 ∈ Proset )
121	119, 120	mp1i 14	. . 3 ⊢ (𝜑 → 𝑊 ∈ Proset )
122	113, 114, 107, 100, 115, 118, 121	mgcval 33427	. 2 ⊢ (𝜑 → (𝐺(𝑉MGalConn𝑊)𝐻 ↔ ((𝐺:(Base‘𝑉)⟶(Base‘𝑊) ∧ 𝐻:(Base‘𝑊)⟶(Base‘𝑉)) ∧ ∀𝑢 ∈ (Base‘𝑉)∀𝑣 ∈ (Base‘𝑊)((𝐺‘𝑢)(le‘𝑊)𝑣 ↔ 𝑢(le‘𝑉)(𝐻‘𝑣)))))
123	29, 112, 122	mpbir2and 726	1 ⊢ (𝜑 → 𝐺(𝑉MGalConn𝑊)𝐻)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  {crab 3412  Vcvv 3450   ⊆ wss 3899  𝒫 cpw 4557  {csn 4584   class class class wbr 5103   ↦ cmpt 5186  ^◡ccnv 5654   “ cima 5658  Fun wfun 6527   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  (class class class)co 7413  Basecbs 17301  lecple 17349   Proset cproset 18380  Posetcpo 18395  toInccipo 18615  MGalConncmgc 33419

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-er 8696  df-map 8828  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-nn 12258  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12529  df-z 12616  df-dec 12737  df-uz 12888  df-fz 13562  df-struct 17239  df-slot 17274  df-ndx 17286  df-base 17302  df-tset 17361  df-ple 17362  df-ocomp 17363  df-proset 18382  df-poset 18401  df-ipo 18616  df-mgc 33421

This theorem is used by: (None)