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Theorem pwrssmgc 30691
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 ↦ (𝐹𝑛))
pwrssmgc.2 𝐻 = (𝑚 ∈ 𝒫 𝑋 ↦ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑚})
pwrssmgc.3 𝑉 = (toInc‘𝒫 𝑌)
pwrssmgc.4 𝑊 = (toInc‘𝒫 𝑋)
pwrssmgc.5 (𝜑𝑋𝐴)
pwrssmgc.6 (𝜑𝑌𝐵)
pwrssmgc.7 (𝜑𝐹:𝑋𝑌)
Assertion
Ref Expression
pwrssmgc (𝜑𝐺(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.
StepHypRef Expression
1 pwrssmgc.5 . . . . . . 7 (𝜑𝑋𝐴)
21adantr 484 . . . . . 6 ((𝜑𝑛 ∈ 𝒫 𝑌) → 𝑋𝐴)
3 cnvimass 5936 . . . . . . . 8 (𝐹𝑛) ⊆ dom 𝐹
4 pwrssmgc.7 . . . . . . . 8 (𝜑𝐹:𝑋𝑌)
53, 4fssdm 6520 . . . . . . 7 (𝜑 → (𝐹𝑛) ⊆ 𝑋)
65adantr 484 . . . . . 6 ((𝜑𝑛 ∈ 𝒫 𝑌) → (𝐹𝑛) ⊆ 𝑋)
72, 6sselpwd 5216 . . . . 5 ((𝜑𝑛 ∈ 𝒫 𝑌) → (𝐹𝑛) ∈ 𝒫 𝑋)
8 pwrssmgc.1 . . . . 5 𝐺 = (𝑛 ∈ 𝒫 𝑌 ↦ (𝐹𝑛))
97, 8fmptd 6869 . . . 4 (𝜑𝐺:𝒫 𝑌⟶𝒫 𝑋)
10 pwrssmgc.6 . . . . . 6 (𝜑𝑌𝐵)
11 pwexg 5266 . . . . . 6 (𝑌𝐵 → 𝒫 𝑌 ∈ V)
12 pwrssmgc.3 . . . . . . 7 𝑉 = (toInc‘𝒫 𝑌)
1312ipobas 17765 . . . . . 6 (𝒫 𝑌 ∈ V → 𝒫 𝑌 = (Base‘𝑉))
1410, 11, 133syl 18 . . . . 5 (𝜑 → 𝒫 𝑌 = (Base‘𝑉))
15 pwexg 5266 . . . . . 6 (𝑋𝐴 → 𝒫 𝑋 ∈ V)
16 pwrssmgc.4 . . . . . . 7 𝑊 = (toInc‘𝒫 𝑋)
1716ipobas 17765 . . . . . 6 (𝒫 𝑋 ∈ V → 𝒫 𝑋 = (Base‘𝑊))
181, 15, 173syl 18 . . . . 5 (𝜑 → 𝒫 𝑋 = (Base‘𝑊))
1914, 18feq23d 6498 . . . 4 (𝜑 → (𝐺:𝒫 𝑌⟶𝒫 𝑋𝐺:(Base‘𝑉)⟶(Base‘𝑊)))
209, 19mpbid 235 . . 3 (𝜑𝐺:(Base‘𝑉)⟶(Base‘𝑊))
2110adantr 484 . . . . . 6 ((𝜑𝑚 ∈ 𝒫 𝑋) → 𝑌𝐵)
22 ssrab2 4042 . . . . . . 7 {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑚} ⊆ 𝑌
2322a1i 11 . . . . . 6 ((𝜑𝑚 ∈ 𝒫 𝑋) → {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑚} ⊆ 𝑌)
2421, 23sselpwd 5216 . . . . 5 ((𝜑𝑚 ∈ 𝒫 𝑋) → {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑚} ∈ 𝒫 𝑌)
25 pwrssmgc.2 . . . . 5 𝐻 = (𝑚 ∈ 𝒫 𝑋 ↦ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑚})
2624, 25fmptd 6869 . . . 4 (𝜑𝐻:𝒫 𝑋⟶𝒫 𝑌)
2718, 14feq23d 6498 . . . 4 (𝜑 → (𝐻:𝒫 𝑋⟶𝒫 𝑌𝐻:(Base‘𝑊)⟶(Base‘𝑉)))
2826, 27mpbid 235 . . 3 (𝜑𝐻:(Base‘𝑊)⟶(Base‘𝑉))
2920, 28jca 515 . 2 (𝜑 → (𝐺:(Base‘𝑉)⟶(Base‘𝑊) ∧ 𝐻:(Base‘𝑊)⟶(Base‘𝑉)))
30 sneq 4560 . . . . . . . . . . . 12 (𝑦 = 𝑗 → {𝑦} = {𝑗})
3130imaeq2d 5916 . . . . . . . . . . 11 (𝑦 = 𝑗 → (𝐹 “ {𝑦}) = (𝐹 “ {𝑗}))
3231sseq1d 3984 . . . . . . . . . 10 (𝑦 = 𝑗 → ((𝐹 “ {𝑦}) ⊆ 𝑣 ↔ (𝐹 “ {𝑗}) ⊆ 𝑣))
33 simplr 768 . . . . . . . . . . . . . 14 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑢 ∈ (Base‘𝑉))
3414ad2antrr 725 . . . . . . . . . . . . . 14 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝒫 𝑌 = (Base‘𝑉))
3533, 34eleqtrrd 2919 . . . . . . . . . . . . 13 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑢 ∈ 𝒫 𝑌)
3635adantr 484 . . . . . . . . . . . 12 ((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (𝐹𝑢) ⊆ 𝑣) → 𝑢 ∈ 𝒫 𝑌)
3736elpwid 4533 . . . . . . . . . . 11 ((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (𝐹𝑢) ⊆ 𝑣) → 𝑢𝑌)
3837sselda 3953 . . . . . . . . . 10 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (𝐹𝑢) ⊆ 𝑣) ∧ 𝑗𝑢) → 𝑗𝑌)
394ffund 6507 . . . . . . . . . . . . 13 (𝜑 → Fun 𝐹)
4039ad4antr 731 . . . . . . . . . . . 12 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (𝐹𝑢) ⊆ 𝑣) ∧ 𝑗𝑢) → Fun 𝐹)
41 snssi 4725 . . . . . . . . . . . . 13 (𝑗𝑢 → {𝑗} ⊆ 𝑢)
4241adantl 485 . . . . . . . . . . . 12 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (𝐹𝑢) ⊆ 𝑣) ∧ 𝑗𝑢) → {𝑗} ⊆ 𝑢)
43 sspreima 30403 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ {𝑗} ⊆ 𝑢) → (𝐹 “ {𝑗}) ⊆ (𝐹𝑢))
4440, 42, 43syl2anc 587 . . . . . . . . . . 11 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (𝐹𝑢) ⊆ 𝑣) ∧ 𝑗𝑢) → (𝐹 “ {𝑗}) ⊆ (𝐹𝑢))
45 simplr 768 . . . . . . . . . . 11 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (𝐹𝑢) ⊆ 𝑣) ∧ 𝑗𝑢) → (𝐹𝑢) ⊆ 𝑣)
4644, 45sstrd 3963 . . . . . . . . . 10 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (𝐹𝑢) ⊆ 𝑣) ∧ 𝑗𝑢) → (𝐹 “ {𝑗}) ⊆ 𝑣)
4732, 38, 46elrabd 3668 . . . . . . . . 9 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (𝐹𝑢) ⊆ 𝑣) ∧ 𝑗𝑢) → 𝑗 ∈ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣})
4847ex 416 . . . . . . . 8 ((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (𝐹𝑢) ⊆ 𝑣) → (𝑗𝑢𝑗 ∈ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}))
4948ssrdv 3959 . . . . . . 7 ((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (𝐹𝑢) ⊆ 𝑣) → 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣})
50 simplr 768 . . . . . . . . . . . 12 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (𝐹𝑢)) → 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣})
514ffnd 6504 . . . . . . . . . . . . . . 15 (𝜑𝐹 Fn 𝑋)
5251ad4antr 731 . . . . . . . . . . . . . 14 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (𝐹𝑢)) → 𝐹 Fn 𝑋)
53 simpr 488 . . . . . . . . . . . . . 14 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (𝐹𝑢)) → 𝑖 ∈ (𝐹𝑢))
54 elpreima 6819 . . . . . . . . . . . . . . 15 (𝐹 Fn 𝑋 → (𝑖 ∈ (𝐹𝑢) ↔ (𝑖𝑋 ∧ (𝐹𝑖) ∈ 𝑢)))
5554biimpa 480 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝑋𝑖 ∈ (𝐹𝑢)) → (𝑖𝑋 ∧ (𝐹𝑖) ∈ 𝑢))
5652, 53, 55syl2anc 587 . . . . . . . . . . . . 13 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (𝐹𝑢)) → (𝑖𝑋 ∧ (𝐹𝑖) ∈ 𝑢))
5756simprd 499 . . . . . . . . . . . 12 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (𝐹𝑢)) → (𝐹𝑖) ∈ 𝑢)
5850, 57sseldd 3954 . . . . . . . . . . 11 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (𝐹𝑢)) → (𝐹𝑖) ∈ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣})
59 sneq 4560 . . . . . . . . . . . . . . 15 (𝑦 = (𝐹𝑖) → {𝑦} = {(𝐹𝑖)})
6059imaeq2d 5916 . . . . . . . . . . . . . 14 (𝑦 = (𝐹𝑖) → (𝐹 “ {𝑦}) = (𝐹 “ {(𝐹𝑖)}))
6160sseq1d 3984 . . . . . . . . . . . . 13 (𝑦 = (𝐹𝑖) → ((𝐹 “ {𝑦}) ⊆ 𝑣 ↔ (𝐹 “ {(𝐹𝑖)}) ⊆ 𝑣))
6261elrab 3666 . . . . . . . . . . . 12 ((𝐹𝑖) ∈ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣} ↔ ((𝐹𝑖) ∈ 𝑌 ∧ (𝐹 “ {(𝐹𝑖)}) ⊆ 𝑣))
6362simprbi 500 . . . . . . . . . . 11 ((𝐹𝑖) ∈ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣} → (𝐹 “ {(𝐹𝑖)}) ⊆ 𝑣)
6458, 63syl 17 . . . . . . . . . 10 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (𝐹𝑢)) → (𝐹 “ {(𝐹𝑖)}) ⊆ 𝑣)
6556simpld 498 . . . . . . . . . . 11 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (𝐹𝑢)) → 𝑖𝑋)
66 eqidd 2825 . . . . . . . . . . 11 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (𝐹𝑢)) → (𝐹𝑖) = (𝐹𝑖))
67 fniniseg 6821 . . . . . . . . . . . 12 (𝐹 Fn 𝑋 → (𝑖 ∈ (𝐹 “ {(𝐹𝑖)}) ↔ (𝑖𝑋 ∧ (𝐹𝑖) = (𝐹𝑖))))
6867biimpar 481 . . . . . . . . . . 11 ((𝐹 Fn 𝑋 ∧ (𝑖𝑋 ∧ (𝐹𝑖) = (𝐹𝑖))) → 𝑖 ∈ (𝐹 “ {(𝐹𝑖)}))
6952, 65, 66, 68syl12anc 835 . . . . . . . . . 10 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (𝐹𝑢)) → 𝑖 ∈ (𝐹 “ {(𝐹𝑖)}))
7064, 69sseldd 3954 . . . . . . . . 9 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (𝐹𝑢)) → 𝑖𝑣)
7170ex 416 . . . . . . . 8 ((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}) → (𝑖 ∈ (𝐹𝑢) → 𝑖𝑣))
7271ssrdv 3959 . . . . . . 7 ((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}) → (𝐹𝑢) ⊆ 𝑣)
7349, 72impbida 800 . . . . . 6 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → ((𝐹𝑢) ⊆ 𝑣𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}))
74 simpr 488 . . . . . . . . 9 ((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑛 = 𝑢) → 𝑛 = 𝑢)
7574imaeq2d 5916 . . . . . . . 8 ((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑛 = 𝑢) → (𝐹𝑛) = (𝐹𝑢))
76 fex 6980 . . . . . . . . . . 11 ((𝐹:𝑋𝑌𝑋𝐴) → 𝐹 ∈ V)
774, 1, 76syl2anc 587 . . . . . . . . . 10 (𝜑𝐹 ∈ V)
78 cnvexg 7624 . . . . . . . . . 10 (𝐹 ∈ V → 𝐹 ∈ V)
79 imaexg 7615 . . . . . . . . . 10 (𝐹 ∈ V → (𝐹𝑢) ∈ V)
8077, 78, 793syl 18 . . . . . . . . 9 (𝜑 → (𝐹𝑢) ∈ V)
8180ad2antrr 725 . . . . . . . 8 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝐹𝑢) ∈ V)
828, 75, 35, 81fvmptd2 6767 . . . . . . 7 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝐺𝑢) = (𝐹𝑢))
8382sseq1d 3984 . . . . . 6 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → ((𝐺𝑢) ⊆ 𝑣 ↔ (𝐹𝑢) ⊆ 𝑣))
84 simpr 488 . . . . . . . . . 10 ((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑚 = 𝑣) → 𝑚 = 𝑣)
8584sseq2d 3985 . . . . . . . . 9 ((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑚 = 𝑣) → ((𝐹 “ {𝑦}) ⊆ 𝑚 ↔ (𝐹 “ {𝑦}) ⊆ 𝑣))
8685rabbidv 3465 . . . . . . . 8 ((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑚 = 𝑣) → {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑚} = {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣})
87 simpr 488 . . . . . . . . 9 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑣 ∈ (Base‘𝑊))
881, 15syl 17 . . . . . . . . . . 11 (𝜑 → 𝒫 𝑋 ∈ V)
8988ad2antrr 725 . . . . . . . . . 10 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝒫 𝑋 ∈ V)
9089, 17syl 17 . . . . . . . . 9 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝒫 𝑋 = (Base‘𝑊))
9187, 90eleqtrrd 2919 . . . . . . . 8 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑣 ∈ 𝒫 𝑋)
9210ad2antrr 725 . . . . . . . . 9 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑌𝐵)
93 ssrab2 4042 . . . . . . . . . 10 {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣} ⊆ 𝑌
9493a1i 11 . . . . . . . . 9 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣} ⊆ 𝑌)
9592, 94sselpwd 5216 . . . . . . . 8 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣} ∈ 𝒫 𝑌)
9625, 86, 91, 95fvmptd2 6767 . . . . . . 7 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝐻𝑣) = {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣})
9796sseq2d 3985 . . . . . 6 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝑢 ⊆ (𝐻𝑣) ↔ 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}))
9873, 83, 973bitr4d 314 . . . . 5 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → ((𝐺𝑢) ⊆ 𝑣𝑢 ⊆ (𝐻𝑣)))
999ad2antrr 725 . . . . . . 7 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝐺:𝒫 𝑌⟶𝒫 𝑋)
10099, 35ffvelrnd 6843 . . . . . 6 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝐺𝑢) ∈ 𝒫 𝑋)
101 eqid 2824 . . . . . . 7 (le‘𝑊) = (le‘𝑊)
10216, 101ipole 17768 . . . . . 6 ((𝒫 𝑋 ∈ V ∧ (𝐺𝑢) ∈ 𝒫 𝑋𝑣 ∈ 𝒫 𝑋) → ((𝐺𝑢)(le‘𝑊)𝑣 ↔ (𝐺𝑢) ⊆ 𝑣))
10389, 100, 91, 102syl3anc 1368 . . . . 5 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → ((𝐺𝑢)(le‘𝑊)𝑣 ↔ (𝐺𝑢) ⊆ 𝑣))
10410, 11syl 17 . . . . . . 7 (𝜑 → 𝒫 𝑌 ∈ V)
105104ad2antrr 725 . . . . . 6 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝒫 𝑌 ∈ V)
10626ad2antrr 725 . . . . . . 7 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝐻:𝒫 𝑋⟶𝒫 𝑌)
107106, 91ffvelrnd 6843 . . . . . 6 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝐻𝑣) ∈ 𝒫 𝑌)
108 eqid 2824 . . . . . . 7 (le‘𝑉) = (le‘𝑉)
10912, 108ipole 17768 . . . . . 6 ((𝒫 𝑌 ∈ V ∧ 𝑢 ∈ 𝒫 𝑌 ∧ (𝐻𝑣) ∈ 𝒫 𝑌) → (𝑢(le‘𝑉)(𝐻𝑣) ↔ 𝑢 ⊆ (𝐻𝑣)))
110105, 35, 107, 109syl3anc 1368 . . . . 5 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝑢(le‘𝑉)(𝐻𝑣) ↔ 𝑢 ⊆ (𝐻𝑣)))
11198, 103, 1103bitr4d 314 . . . 4 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → ((𝐺𝑢)(le‘𝑊)𝑣𝑢(le‘𝑉)(𝐻𝑣)))
112111anasss 470 . . 3 ((𝜑 ∧ (𝑢 ∈ (Base‘𝑉) ∧ 𝑣 ∈ (Base‘𝑊))) → ((𝐺𝑢)(le‘𝑊)𝑣𝑢(le‘𝑉)(𝐻𝑣)))
113112ralrimivva 3186 . 2 (𝜑 → ∀𝑢 ∈ (Base‘𝑉)∀𝑣 ∈ (Base‘𝑊)((𝐺𝑢)(le‘𝑊)𝑣𝑢(le‘𝑉)(𝐻𝑣)))
114 eqid 2824 . . 3 (Base‘𝑉) = (Base‘𝑉)
115 eqid 2824 . . 3 (Base‘𝑊) = (Base‘𝑊)
116 eqid 2824 . . 3 (𝑉MGalConn𝑊) = (𝑉MGalConn𝑊)
11712ipopos 17770 . . . 4 𝑉 ∈ Poset
118 posprs 17559 . . . 4 (𝑉 ∈ Poset → 𝑉 ∈ Proset )
119117, 118mp1i 13 . . 3 (𝜑𝑉 ∈ Proset )
12016ipopos 17770 . . . 4 𝑊 ∈ Poset
121 posprs 17559 . . . 4 (𝑊 ∈ Poset → 𝑊 ∈ Proset )
122120, 121mp1i 13 . . 3 (𝜑𝑊 ∈ Proset )
123114, 115, 108, 101, 116, 119, 122mgcval 30680 . 2 (𝜑 → (𝐺(𝑉MGalConn𝑊)𝐻 ↔ ((𝐺:(Base‘𝑉)⟶(Base‘𝑊) ∧ 𝐻:(Base‘𝑊)⟶(Base‘𝑉)) ∧ ∀𝑢 ∈ (Base‘𝑉)∀𝑣 ∈ (Base‘𝑊)((𝐺𝑢)(le‘𝑊)𝑣𝑢(le‘𝑉)(𝐻𝑣)))))
12429, 113, 123mpbir2and 712 1 (𝜑𝐺(𝑉MGalConn𝑊)𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3133  {crab 3137  Vcvv 3480  wss 3919  𝒫 cpw 4522  {csn 4550   class class class wbr 5052  cmpt 5132  ccnv 5541  cima 5545  Fun wfun 6337   Fn wfn 6338  wf 6339  cfv 6343  (class class class)co 7149  Basecbs 16483  lecple 16572   Proset cproset 17536  Posetcpo 17550  toInccipo 17761  MGalConncmgc 30672
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-map 8404  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-2 11697  df-3 11698  df-4 11699  df-5 11700  df-6 11701  df-7 11702  df-8 11703  df-9 11704  df-n0 11895  df-z 11979  df-dec 12096  df-uz 12241  df-fz 12895  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-tset 16584  df-ple 16585  df-ocomp 16586  df-proset 17538  df-poset 17556  df-ipo 17762  df-mgc 30674
This theorem is referenced by: (None)
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