Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pwrssmgc Structured version   Visualization version   GIF version

 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})
Assertion
Ref Expression
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)

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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