Theorem pwrssmgc 32985

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 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝒫 𝑌) → 𝑋 ∈ 𝐴)
3		cnvimass 6074	. . . . . . . 8 ⊢ (◡𝐹 “ 𝑛) ⊆ dom 𝐹
4		pwrssmgc.7	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)
5	3, 4	fssdm 6730	. . . . . . 7 ⊢ (𝜑 → (◡𝐹 “ 𝑛) ⊆ 𝑋)
6	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝒫 𝑌) → (◡𝐹 “ 𝑛) ⊆ 𝑋)
7	2, 6	sselpwd 5303	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝒫 𝑌) → (◡𝐹 “ 𝑛) ∈ 𝒫 𝑋)
8		pwrssmgc.1	. . . . 5 ⊢ 𝐺 = (𝑛 ∈ 𝒫 𝑌 ↦ (◡𝐹 “ 𝑛))
9	7, 8	fmptd 7109	. . . 4 ⊢ (𝜑 → 𝐺:𝒫 𝑌⟶𝒫 𝑋)
10		pwrssmgc.6	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
11		pwexg 5353	. . . . . 6 ⊢ (𝑌 ∈ 𝐵 → 𝒫 𝑌 ∈ V)
12		pwrssmgc.3	. . . . . . 7 ⊢ 𝑉 = (toInc‘𝒫 𝑌)
13	12	ipobas 18546	. . . . . 6 ⊢ (𝒫 𝑌 ∈ V → 𝒫 𝑌 = (Base‘𝑉))
14	10, 11, 13	3syl 18	. . . . 5 ⊢ (𝜑 → 𝒫 𝑌 = (Base‘𝑉))
15		pwexg 5353	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → 𝒫 𝑋 ∈ V)
16		pwrssmgc.4	. . . . . . 7 ⊢ 𝑊 = (toInc‘𝒫 𝑋)
17	16	ipobas 18546	. . . . . 6 ⊢ (𝒫 𝑋 ∈ V → 𝒫 𝑋 = (Base‘𝑊))
18	1, 15, 17	3syl 18	. . . . 5 ⊢ (𝜑 → 𝒫 𝑋 = (Base‘𝑊))
19	14, 18	feq23d 6706	. . . 4 ⊢ (𝜑 → (𝐺:𝒫 𝑌⟶𝒫 𝑋 ↔ 𝐺:(Base‘𝑉)⟶(Base‘𝑊)))
20	9, 19	mpbid 232	. . 3 ⊢ (𝜑 → 𝐺:(Base‘𝑉)⟶(Base‘𝑊))
21	10	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝒫 𝑋) → 𝑌 ∈ 𝐵)
22		ssrab2 4060	. . . . . . 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 5303	. . . . 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 7109	. . . 4 ⊢ (𝜑 → 𝐻:𝒫 𝑋⟶𝒫 𝑌)
27	18, 14	feq23d 6706	. . . 4 ⊢ (𝜑 → (𝐻:𝒫 𝑋⟶𝒫 𝑌 ↔ 𝐻:(Base‘𝑊)⟶(Base‘𝑉)))
28	26, 27	mpbid 232	. . 3 ⊢ (𝜑 → 𝐻:(Base‘𝑊)⟶(Base‘𝑉))
29	20, 28	jca 511	. 2 ⊢ (𝜑 → (𝐺:(Base‘𝑉)⟶(Base‘𝑊) ∧ 𝐻:(Base‘𝑊)⟶(Base‘𝑉)))
30		sneq 4616	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑗 → {𝑦} = {𝑗})
31	30	imaeq2d 6052	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑗 → (◡𝐹 “ {𝑦}) = (◡𝐹 “ {𝑗}))
32	31	sseq1d 3995	. . . . . . . . . 10 ⊢ (𝑦 = 𝑗 → ((◡𝐹 “ {𝑦}) ⊆ 𝑣 ↔ (◡𝐹 “ {𝑗}) ⊆ 𝑣))
33		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑢 ∈ (Base‘𝑉))
34	14	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝒫 𝑌 = (Base‘𝑉))
35	33, 34	eleqtrrd 2838	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑢 ∈ 𝒫 𝑌)
36	35	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (◡𝐹 “ 𝑢) ⊆ 𝑣) → 𝑢 ∈ 𝒫 𝑌)
37	36	elpwid 4589	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (◡𝐹 “ 𝑢) ⊆ 𝑣) → 𝑢 ⊆ 𝑌)
38	37	sselda 3963	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (◡𝐹 “ 𝑢) ⊆ 𝑣) ∧ 𝑗 ∈ 𝑢) → 𝑗 ∈ 𝑌)
39	4	ffund 6715	. . . . . . . . . . . . 13 ⊢ (𝜑 → Fun 𝐹)
40	39	ad4antr 732	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (◡𝐹 “ 𝑢) ⊆ 𝑣) ∧ 𝑗 ∈ 𝑢) → Fun 𝐹)
41		snssi 4789	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ 𝑢 → {𝑗} ⊆ 𝑢)
42	41	adantl 481	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (◡𝐹 “ 𝑢) ⊆ 𝑣) ∧ 𝑗 ∈ 𝑢) → {𝑗} ⊆ 𝑢)
43		sspreima 7063	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ {𝑗} ⊆ 𝑢) → (◡𝐹 “ {𝑗}) ⊆ (◡𝐹 “ 𝑢))
44	40, 42, 43	syl2anc 584	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (◡𝐹 “ 𝑢) ⊆ 𝑣) ∧ 𝑗 ∈ 𝑢) → (◡𝐹 “ {𝑗}) ⊆ (◡𝐹 “ 𝑢))
45		simplr 768	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (◡𝐹 “ 𝑢) ⊆ 𝑣) ∧ 𝑗 ∈ 𝑢) → (◡𝐹 “ 𝑢) ⊆ 𝑣)
46	44, 45	sstrd 3974	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (◡𝐹 “ 𝑢) ⊆ 𝑣) ∧ 𝑗 ∈ 𝑢) → (◡𝐹 “ {𝑗}) ⊆ 𝑣)
47	32, 38, 46	elrabd 3678	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (◡𝐹 “ 𝑢) ⊆ 𝑣) ∧ 𝑗 ∈ 𝑢) → 𝑗 ∈ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣})
48	47	ex 412	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (◡𝐹 “ 𝑢) ⊆ 𝑣) → (𝑗 ∈ 𝑢 → 𝑗 ∈ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣}))
49	48	ssrdv 3969	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (◡𝐹 “ 𝑢) ⊆ 𝑣) → 𝑢 ⊆ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣})
50		simplr 768	. . . . . . . . . . . 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 6712	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 Fn 𝑋)
52	51	ad4antr 732	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (◡𝐹 “ 𝑢)) → 𝐹 Fn 𝑋)
53		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (◡𝐹 “ 𝑢)) → 𝑖 ∈ (◡𝐹 “ 𝑢))
54		elpreima 7053	. . . . . . . . . . . . . . 15 ⊢ (𝐹 Fn 𝑋 → (𝑖 ∈ (◡𝐹 “ 𝑢) ↔ (𝑖 ∈ 𝑋 ∧ (𝐹‘𝑖) ∈ 𝑢)))
55	54	biimpa 476	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑖 ∈ (◡𝐹 “ 𝑢)) → (𝑖 ∈ 𝑋 ∧ (𝐹‘𝑖) ∈ 𝑢))
56	52, 53, 55	syl2anc 584	. . . . . . . . . . . . 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 495	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (◡𝐹 “ 𝑢)) → (𝐹‘𝑖) ∈ 𝑢)
58	50, 57	sseldd 3964	. . . . . . . . . . 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 4616	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝐹‘𝑖) → {𝑦} = {(𝐹‘𝑖)})
60	59	imaeq2d 6052	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐹‘𝑖) → (◡𝐹 “ {𝑦}) = (◡𝐹 “ {(𝐹‘𝑖)}))
61	60	sseq1d 3995	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐹‘𝑖) → ((◡𝐹 “ {𝑦}) ⊆ 𝑣 ↔ (◡𝐹 “ {(𝐹‘𝑖)}) ⊆ 𝑣))
62	61	elrab 3676	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑖) ∈ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣} ↔ ((𝐹‘𝑖) ∈ 𝑌 ∧ (◡𝐹 “ {(𝐹‘𝑖)}) ⊆ 𝑣))
63	62	simprbi 496	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑖) ∈ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣} → (◡𝐹 “ {(𝐹‘𝑖)}) ⊆ 𝑣)
64	58, 63	syl 17	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (◡𝐹 “ 𝑢)) → (◡𝐹 “ {(𝐹‘𝑖)}) ⊆ 𝑣)
65	56	simpld 494	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (◡𝐹 “ 𝑢)) → 𝑖 ∈ 𝑋)
66		eqidd 2737	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (◡𝐹 “ 𝑢)) → (𝐹‘𝑖) = (𝐹‘𝑖))
67		fniniseg 7055	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝑋 → (𝑖 ∈ (◡𝐹 “ {(𝐹‘𝑖)}) ↔ (𝑖 ∈ 𝑋 ∧ (𝐹‘𝑖) = (𝐹‘𝑖))))
68	67	biimpar 477	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝑋 ∧ (𝑖 ∈ 𝑋 ∧ (𝐹‘𝑖) = (𝐹‘𝑖))) → 𝑖 ∈ (◡𝐹 “ {(𝐹‘𝑖)}))
69	52, 65, 66, 68	syl12anc 836	. . . . . . . . . 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 3964	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (◡𝐹 “ 𝑢)) → 𝑖 ∈ 𝑣)
71	70	ex 412	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣}) → (𝑖 ∈ (◡𝐹 “ 𝑢) → 𝑖 ∈ 𝑣))
72	71	ssrdv 3969	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣}) → (◡𝐹 “ 𝑢) ⊆ 𝑣)
73	49, 72	impbida 800	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → ((◡𝐹 “ 𝑢) ⊆ 𝑣 ↔ 𝑢 ⊆ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣}))
74		simpr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑛 = 𝑢) → 𝑛 = 𝑢)
75	74	imaeq2d 6052	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑛 = 𝑢) → (◡𝐹 “ 𝑛) = (◡𝐹 “ 𝑢))
76	4, 1	fexd 7224	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ V)
77		cnvexg 7925	. . . . . . . . . 10 ⊢ (𝐹 ∈ V → ◡𝐹 ∈ V)
78		imaexg 7914	. . . . . . . . . 10 ⊢ (◡𝐹 ∈ V → (◡𝐹 “ 𝑢) ∈ V)
79	76, 77, 78	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → (◡𝐹 “ 𝑢) ∈ V)
80	79	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → (◡𝐹 “ 𝑢) ∈ V)
81	8, 75, 35, 80	fvmptd2 6999	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝐺‘𝑢) = (◡𝐹 “ 𝑢))
82	81	sseq1d 3995	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → ((𝐺‘𝑢) ⊆ 𝑣 ↔ (◡𝐹 “ 𝑢) ⊆ 𝑣))
83		simpr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑚 = 𝑣) → 𝑚 = 𝑣)
84	83	sseq2d 3996	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑚 = 𝑣) → ((◡𝐹 “ {𝑦}) ⊆ 𝑚 ↔ (◡𝐹 “ {𝑦}) ⊆ 𝑣))
85	84	rabbidv 3428	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑚 = 𝑣) → {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑚} = {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣})
86		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑣 ∈ (Base‘𝑊))
87	1, 15	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝒫 𝑋 ∈ V)
88	87	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝒫 𝑋 ∈ V)
89	88, 17	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝒫 𝑋 = (Base‘𝑊))
90	86, 89	eleqtrrd 2838	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑣 ∈ 𝒫 𝑋)
91	10	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑌 ∈ 𝐵)
92		ssrab2 4060	. . . . . . . . . 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 5303	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣} ∈ 𝒫 𝑌)
95	25, 85, 90, 94	fvmptd2 6999	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝐻‘𝑣) = {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣})
96	95	sseq2d 3996	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝑢 ⊆ (𝐻‘𝑣) ↔ 𝑢 ⊆ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑣}))
97	73, 82, 96	3bitr4d 311	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → ((𝐺‘𝑢) ⊆ 𝑣 ↔ 𝑢 ⊆ (𝐻‘𝑣)))
98	9	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝐺:𝒫 𝑌⟶𝒫 𝑋)
99	98, 35	ffvelcdmd 7080	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝐺‘𝑢) ∈ 𝒫 𝑋)
100		eqid 2736	. . . . . . 7 ⊢ (le‘𝑊) = (le‘𝑊)
101	16, 100	ipole 18549	. . . . . 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 1373	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → ((𝐺‘𝑢)(le‘𝑊)𝑣 ↔ (𝐺‘𝑢) ⊆ 𝑣))
103	10, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝒫 𝑌 ∈ V)
104	103	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝒫 𝑌 ∈ V)
105	26	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝐻:𝒫 𝑋⟶𝒫 𝑌)
106	105, 90	ffvelcdmd 7080	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝐻‘𝑣) ∈ 𝒫 𝑌)
107		eqid 2736	. . . . . . 7 ⊢ (le‘𝑉) = (le‘𝑉)
108	12, 107	ipole 18549	. . . . . 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 1373	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝑢(le‘𝑉)(𝐻‘𝑣) ↔ 𝑢 ⊆ (𝐻‘𝑣)))
110	97, 102, 109	3bitr4d 311	. . . 4 ⊢ (((𝜑 ∧ 𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → ((𝐺‘𝑢)(le‘𝑊)𝑣 ↔ 𝑢(le‘𝑉)(𝐻‘𝑣)))
111	110	anasss 466	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑉) ∧ 𝑣 ∈ (Base‘𝑊))) → ((𝐺‘𝑢)(le‘𝑊)𝑣 ↔ 𝑢(le‘𝑉)(𝐻‘𝑣)))
112	111	ralrimivva 3188	. 2 ⊢ (𝜑 → ∀𝑢 ∈ (Base‘𝑉)∀𝑣 ∈ (Base‘𝑊)((𝐺‘𝑢)(le‘𝑊)𝑣 ↔ 𝑢(le‘𝑉)(𝐻‘𝑣)))
113		eqid 2736	. . 3 ⊢ (Base‘𝑉) = (Base‘𝑉)
114		eqid 2736	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
115		eqid 2736	. . 3 ⊢ (𝑉MGalConn𝑊) = (𝑉MGalConn𝑊)
116	12	ipopos 18551	. . . 4 ⊢ 𝑉 ∈ Poset
117		posprs 18333	. . . 4 ⊢ (𝑉 ∈ Poset → 𝑉 ∈ Proset )
118	116, 117	mp1i 13	. . 3 ⊢ (𝜑 → 𝑉 ∈ Proset )
119	16	ipopos 18551	. . . 4 ⊢ 𝑊 ∈ Poset
120		posprs 18333	. . . 4 ⊢ (𝑊 ∈ Poset → 𝑊 ∈ Proset )
121	119, 120	mp1i 13	. . 3 ⊢ (𝜑 → 𝑊 ∈ Proset )
122	113, 114, 107, 100, 115, 118, 121	mgcval 32972	. 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 713	1 ⊢ (𝜑 → 𝐺(𝑉MGalConn𝑊)𝐻)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3052  {crab 3420  Vcvv 3464   ⊆ wss 3931  𝒫 cpw 4580  {csn 4606   class class class wbr 5124   ↦ cmpt 5206  ^◡ccnv 5658   “ cima 5662  Fun wfun 6530   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  lecple 17283   Proset cproset 18309  Posetcpo 18324  toInccipo 18542  MGalConncmgc 32964

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8724  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12507  df-z 12594  df-dec 12714  df-uz 12858  df-fz 13530  df-struct 17171  df-slot 17206  df-ndx 17218  df-base 17234  df-tset 17295  df-ple 17296  df-ocomp 17297  df-proset 18311  df-poset 18330  df-ipo 18543  df-mgc 32966

This theorem is referenced by: (None)