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Theorem pwrssmgc 31860
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 481 . . . . . 6 ((𝜑𝑛 ∈ 𝒫 𝑌) → 𝑋𝐴)
3 cnvimass 6033 . . . . . . . 8 (𝐹𝑛) ⊆ dom 𝐹
4 pwrssmgc.7 . . . . . . . 8 (𝜑𝐹:𝑋𝑌)
53, 4fssdm 6688 . . . . . . 7 (𝜑 → (𝐹𝑛) ⊆ 𝑋)
65adantr 481 . . . . . 6 ((𝜑𝑛 ∈ 𝒫 𝑌) → (𝐹𝑛) ⊆ 𝑋)
72, 6sselpwd 5283 . . . . 5 ((𝜑𝑛 ∈ 𝒫 𝑌) → (𝐹𝑛) ∈ 𝒫 𝑋)
8 pwrssmgc.1 . . . . 5 𝐺 = (𝑛 ∈ 𝒫 𝑌 ↦ (𝐹𝑛))
97, 8fmptd 7062 . . . 4 (𝜑𝐺:𝒫 𝑌⟶𝒫 𝑋)
10 pwrssmgc.6 . . . . . 6 (𝜑𝑌𝐵)
11 pwexg 5333 . . . . . 6 (𝑌𝐵 → 𝒫 𝑌 ∈ V)
12 pwrssmgc.3 . . . . . . 7 𝑉 = (toInc‘𝒫 𝑌)
1312ipobas 18420 . . . . . 6 (𝒫 𝑌 ∈ V → 𝒫 𝑌 = (Base‘𝑉))
1410, 11, 133syl 18 . . . . 5 (𝜑 → 𝒫 𝑌 = (Base‘𝑉))
15 pwexg 5333 . . . . . 6 (𝑋𝐴 → 𝒫 𝑋 ∈ V)
16 pwrssmgc.4 . . . . . . 7 𝑊 = (toInc‘𝒫 𝑋)
1716ipobas 18420 . . . . . 6 (𝒫 𝑋 ∈ V → 𝒫 𝑋 = (Base‘𝑊))
181, 15, 173syl 18 . . . . 5 (𝜑 → 𝒫 𝑋 = (Base‘𝑊))
1914, 18feq23d 6663 . . . 4 (𝜑 → (𝐺:𝒫 𝑌⟶𝒫 𝑋𝐺:(Base‘𝑉)⟶(Base‘𝑊)))
209, 19mpbid 231 . . 3 (𝜑𝐺:(Base‘𝑉)⟶(Base‘𝑊))
2110adantr 481 . . . . . 6 ((𝜑𝑚 ∈ 𝒫 𝑋) → 𝑌𝐵)
22 ssrab2 4037 . . . . . . 7 {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑚} ⊆ 𝑌
2322a1i 11 . . . . . 6 ((𝜑𝑚 ∈ 𝒫 𝑋) → {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑚} ⊆ 𝑌)
2421, 23sselpwd 5283 . . . . 5 ((𝜑𝑚 ∈ 𝒫 𝑋) → {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑚} ∈ 𝒫 𝑌)
25 pwrssmgc.2 . . . . 5 𝐻 = (𝑚 ∈ 𝒫 𝑋 ↦ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑚})
2624, 25fmptd 7062 . . . 4 (𝜑𝐻:𝒫 𝑋⟶𝒫 𝑌)
2718, 14feq23d 6663 . . . 4 (𝜑 → (𝐻:𝒫 𝑋⟶𝒫 𝑌𝐻:(Base‘𝑊)⟶(Base‘𝑉)))
2826, 27mpbid 231 . . 3 (𝜑𝐻:(Base‘𝑊)⟶(Base‘𝑉))
2920, 28jca 512 . 2 (𝜑 → (𝐺:(Base‘𝑉)⟶(Base‘𝑊) ∧ 𝐻:(Base‘𝑊)⟶(Base‘𝑉)))
30 sneq 4596 . . . . . . . . . . . 12 (𝑦 = 𝑗 → {𝑦} = {𝑗})
3130imaeq2d 6013 . . . . . . . . . . 11 (𝑦 = 𝑗 → (𝐹 “ {𝑦}) = (𝐹 “ {𝑗}))
3231sseq1d 3975 . . . . . . . . . 10 (𝑦 = 𝑗 → ((𝐹 “ {𝑦}) ⊆ 𝑣 ↔ (𝐹 “ {𝑗}) ⊆ 𝑣))
33 simplr 767 . . . . . . . . . . . . . 14 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑢 ∈ (Base‘𝑉))
3414ad2antrr 724 . . . . . . . . . . . . . 14 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝒫 𝑌 = (Base‘𝑉))
3533, 34eleqtrrd 2841 . . . . . . . . . . . . 13 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑢 ∈ 𝒫 𝑌)
3635adantr 481 . . . . . . . . . . . 12 ((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (𝐹𝑢) ⊆ 𝑣) → 𝑢 ∈ 𝒫 𝑌)
3736elpwid 4569 . . . . . . . . . . 11 ((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (𝐹𝑢) ⊆ 𝑣) → 𝑢𝑌)
3837sselda 3944 . . . . . . . . . 10 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (𝐹𝑢) ⊆ 𝑣) ∧ 𝑗𝑢) → 𝑗𝑌)
394ffund 6672 . . . . . . . . . . . . 13 (𝜑 → Fun 𝐹)
4039ad4antr 730 . . . . . . . . . . . 12 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (𝐹𝑢) ⊆ 𝑣) ∧ 𝑗𝑢) → Fun 𝐹)
41 snssi 4768 . . . . . . . . . . . . 13 (𝑗𝑢 → {𝑗} ⊆ 𝑢)
4241adantl 482 . . . . . . . . . . . 12 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (𝐹𝑢) ⊆ 𝑣) ∧ 𝑗𝑢) → {𝑗} ⊆ 𝑢)
43 sspreima 7018 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ {𝑗} ⊆ 𝑢) → (𝐹 “ {𝑗}) ⊆ (𝐹𝑢))
4440, 42, 43syl2anc 584 . . . . . . . . . . 11 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (𝐹𝑢) ⊆ 𝑣) ∧ 𝑗𝑢) → (𝐹 “ {𝑗}) ⊆ (𝐹𝑢))
45 simplr 767 . . . . . . . . . . 11 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (𝐹𝑢) ⊆ 𝑣) ∧ 𝑗𝑢) → (𝐹𝑢) ⊆ 𝑣)
4644, 45sstrd 3954 . . . . . . . . . 10 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (𝐹𝑢) ⊆ 𝑣) ∧ 𝑗𝑢) → (𝐹 “ {𝑗}) ⊆ 𝑣)
4732, 38, 46elrabd 3647 . . . . . . . . 9 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (𝐹𝑢) ⊆ 𝑣) ∧ 𝑗𝑢) → 𝑗 ∈ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣})
4847ex 413 . . . . . . . 8 ((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (𝐹𝑢) ⊆ 𝑣) → (𝑗𝑢𝑗 ∈ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}))
4948ssrdv 3950 . . . . . . 7 ((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ (𝐹𝑢) ⊆ 𝑣) → 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣})
50 simplr 767 . . . . . . . . . . . 12 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (𝐹𝑢)) → 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣})
514ffnd 6669 . . . . . . . . . . . . . . 15 (𝜑𝐹 Fn 𝑋)
5251ad4antr 730 . . . . . . . . . . . . . 14 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (𝐹𝑢)) → 𝐹 Fn 𝑋)
53 simpr 485 . . . . . . . . . . . . . 14 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (𝐹𝑢)) → 𝑖 ∈ (𝐹𝑢))
54 elpreima 7008 . . . . . . . . . . . . . . 15 (𝐹 Fn 𝑋 → (𝑖 ∈ (𝐹𝑢) ↔ (𝑖𝑋 ∧ (𝐹𝑖) ∈ 𝑢)))
5554biimpa 477 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝑋𝑖 ∈ (𝐹𝑢)) → (𝑖𝑋 ∧ (𝐹𝑖) ∈ 𝑢))
5652, 53, 55syl2anc 584 . . . . . . . . . . . . 13 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (𝐹𝑢)) → (𝑖𝑋 ∧ (𝐹𝑖) ∈ 𝑢))
5756simprd 496 . . . . . . . . . . . 12 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (𝐹𝑢)) → (𝐹𝑖) ∈ 𝑢)
5850, 57sseldd 3945 . . . . . . . . . . 11 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (𝐹𝑢)) → (𝐹𝑖) ∈ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣})
59 sneq 4596 . . . . . . . . . . . . . . 15 (𝑦 = (𝐹𝑖) → {𝑦} = {(𝐹𝑖)})
6059imaeq2d 6013 . . . . . . . . . . . . . 14 (𝑦 = (𝐹𝑖) → (𝐹 “ {𝑦}) = (𝐹 “ {(𝐹𝑖)}))
6160sseq1d 3975 . . . . . . . . . . . . 13 (𝑦 = (𝐹𝑖) → ((𝐹 “ {𝑦}) ⊆ 𝑣 ↔ (𝐹 “ {(𝐹𝑖)}) ⊆ 𝑣))
6261elrab 3645 . . . . . . . . . . . 12 ((𝐹𝑖) ∈ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣} ↔ ((𝐹𝑖) ∈ 𝑌 ∧ (𝐹 “ {(𝐹𝑖)}) ⊆ 𝑣))
6362simprbi 497 . . . . . . . . . . 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 495 . . . . . . . . . . 11 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (𝐹𝑢)) → 𝑖𝑋)
66 eqidd 2737 . . . . . . . . . . 11 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (𝐹𝑢)) → (𝐹𝑖) = (𝐹𝑖))
67 fniniseg 7010 . . . . . . . . . . . 12 (𝐹 Fn 𝑋 → (𝑖 ∈ (𝐹 “ {(𝐹𝑖)}) ↔ (𝑖𝑋 ∧ (𝐹𝑖) = (𝐹𝑖))))
6867biimpar 478 . . . . . . . . . . 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 3945 . . . . . . . . 9 (((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}) ∧ 𝑖 ∈ (𝐹𝑢)) → 𝑖𝑣)
7170ex 413 . . . . . . . 8 ((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}) → (𝑖 ∈ (𝐹𝑢) → 𝑖𝑣))
7271ssrdv 3950 . . . . . . 7 ((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}) → (𝐹𝑢) ⊆ 𝑣)
7349, 72impbida 799 . . . . . 6 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → ((𝐹𝑢) ⊆ 𝑣𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}))
74 simpr 485 . . . . . . . . 9 ((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑛 = 𝑢) → 𝑛 = 𝑢)
7574imaeq2d 6013 . . . . . . . 8 ((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑛 = 𝑢) → (𝐹𝑛) = (𝐹𝑢))
764, 1fexd 7177 . . . . . . . . . 10 (𝜑𝐹 ∈ V)
77 cnvexg 7861 . . . . . . . . . 10 (𝐹 ∈ V → 𝐹 ∈ V)
78 imaexg 7852 . . . . . . . . . 10 (𝐹 ∈ V → (𝐹𝑢) ∈ V)
7976, 77, 783syl 18 . . . . . . . . 9 (𝜑 → (𝐹𝑢) ∈ V)
8079ad2antrr 724 . . . . . . . 8 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝐹𝑢) ∈ V)
818, 75, 35, 80fvmptd2 6956 . . . . . . 7 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝐺𝑢) = (𝐹𝑢))
8281sseq1d 3975 . . . . . 6 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → ((𝐺𝑢) ⊆ 𝑣 ↔ (𝐹𝑢) ⊆ 𝑣))
83 simpr 485 . . . . . . . . . 10 ((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑚 = 𝑣) → 𝑚 = 𝑣)
8483sseq2d 3976 . . . . . . . . 9 ((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑚 = 𝑣) → ((𝐹 “ {𝑦}) ⊆ 𝑚 ↔ (𝐹 “ {𝑦}) ⊆ 𝑣))
8584rabbidv 3415 . . . . . . . 8 ((((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑚 = 𝑣) → {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑚} = {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣})
86 simpr 485 . . . . . . . . 9 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑣 ∈ (Base‘𝑊))
871, 15syl 17 . . . . . . . . . . 11 (𝜑 → 𝒫 𝑋 ∈ V)
8887ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝒫 𝑋 ∈ V)
8988, 17syl 17 . . . . . . . . 9 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝒫 𝑋 = (Base‘𝑊))
9086, 89eleqtrrd 2841 . . . . . . . 8 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑣 ∈ 𝒫 𝑋)
9110ad2antrr 724 . . . . . . . . 9 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑌𝐵)
92 ssrab2 4037 . . . . . . . . . 10 {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣} ⊆ 𝑌
9392a1i 11 . . . . . . . . 9 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣} ⊆ 𝑌)
9491, 93sselpwd 5283 . . . . . . . 8 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣} ∈ 𝒫 𝑌)
9525, 85, 90, 94fvmptd2 6956 . . . . . . 7 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝐻𝑣) = {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣})
9695sseq2d 3976 . . . . . 6 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝑢 ⊆ (𝐻𝑣) ↔ 𝑢 ⊆ {𝑦𝑌 ∣ (𝐹 “ {𝑦}) ⊆ 𝑣}))
9773, 82, 963bitr4d 310 . . . . 5 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → ((𝐺𝑢) ⊆ 𝑣𝑢 ⊆ (𝐻𝑣)))
989ad2antrr 724 . . . . . . 7 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝐺:𝒫 𝑌⟶𝒫 𝑋)
9998, 35ffvelcdmd 7036 . . . . . 6 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝐺𝑢) ∈ 𝒫 𝑋)
100 eqid 2736 . . . . . . 7 (le‘𝑊) = (le‘𝑊)
10116, 100ipole 18423 . . . . . 6 ((𝒫 𝑋 ∈ V ∧ (𝐺𝑢) ∈ 𝒫 𝑋𝑣 ∈ 𝒫 𝑋) → ((𝐺𝑢)(le‘𝑊)𝑣 ↔ (𝐺𝑢) ⊆ 𝑣))
10288, 99, 90, 101syl3anc 1371 . . . . 5 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → ((𝐺𝑢)(le‘𝑊)𝑣 ↔ (𝐺𝑢) ⊆ 𝑣))
10310, 11syl 17 . . . . . . 7 (𝜑 → 𝒫 𝑌 ∈ V)
104103ad2antrr 724 . . . . . 6 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝒫 𝑌 ∈ V)
10526ad2antrr 724 . . . . . . 7 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → 𝐻:𝒫 𝑋⟶𝒫 𝑌)
106105, 90ffvelcdmd 7036 . . . . . 6 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝐻𝑣) ∈ 𝒫 𝑌)
107 eqid 2736 . . . . . . 7 (le‘𝑉) = (le‘𝑉)
10812, 107ipole 18423 . . . . . 6 ((𝒫 𝑌 ∈ V ∧ 𝑢 ∈ 𝒫 𝑌 ∧ (𝐻𝑣) ∈ 𝒫 𝑌) → (𝑢(le‘𝑉)(𝐻𝑣) ↔ 𝑢 ⊆ (𝐻𝑣)))
109104, 35, 106, 108syl3anc 1371 . . . . 5 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝑢(le‘𝑉)(𝐻𝑣) ↔ 𝑢 ⊆ (𝐻𝑣)))
11097, 102, 1093bitr4d 310 . . . 4 (((𝜑𝑢 ∈ (Base‘𝑉)) ∧ 𝑣 ∈ (Base‘𝑊)) → ((𝐺𝑢)(le‘𝑊)𝑣𝑢(le‘𝑉)(𝐻𝑣)))
111110anasss 467 . . 3 ((𝜑 ∧ (𝑢 ∈ (Base‘𝑉) ∧ 𝑣 ∈ (Base‘𝑊))) → ((𝐺𝑢)(le‘𝑊)𝑣𝑢(le‘𝑉)(𝐻𝑣)))
112111ralrimivva 3197 . 2 (𝜑 → ∀𝑢 ∈ (Base‘𝑉)∀𝑣 ∈ (Base‘𝑊)((𝐺𝑢)(le‘𝑊)𝑣𝑢(le‘𝑉)(𝐻𝑣)))
113 eqid 2736 . . 3 (Base‘𝑉) = (Base‘𝑉)
114 eqid 2736 . . 3 (Base‘𝑊) = (Base‘𝑊)
115 eqid 2736 . . 3 (𝑉MGalConn𝑊) = (𝑉MGalConn𝑊)
11612ipopos 18425 . . . 4 𝑉 ∈ Poset
117 posprs 18205 . . . 4 (𝑉 ∈ Poset → 𝑉 ∈ Proset )
118116, 117mp1i 13 . . 3 (𝜑𝑉 ∈ Proset )
11916ipopos 18425 . . . 4 𝑊 ∈ Poset
120 posprs 18205 . . . 4 (𝑊 ∈ Poset → 𝑊 ∈ Proset )
121119, 120mp1i 13 . . 3 (𝜑𝑊 ∈ Proset )
122113, 114, 107, 100, 115, 118, 121mgcval 31847 . 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)))))
12329, 112, 122mpbir2and 711 1 (𝜑𝐺(𝑉MGalConn𝑊)𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  {crab 3407  Vcvv 3445  wss 3910  𝒫 cpw 4560  {csn 4586   class class class wbr 5105  cmpt 5188  ccnv 5632  cima 5636  Fun wfun 6490   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  Basecbs 17083  lecple 17140   Proset cproset 18182  Posetcpo 18196  toInccipo 18416  MGalConncmgc 31839
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-struct 17019  df-slot 17054  df-ndx 17066  df-base 17084  df-tset 17152  df-ple 17153  df-ocomp 17154  df-proset 18184  df-poset 18202  df-ipo 18417  df-mgc 31841
This theorem is referenced by: (None)
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