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Theorem mgcoval 31895
Description: Operation value of the monotone Galois connection. (Contributed by Thierry Arnoux, 23-Apr-2024.)
Hypotheses
Ref Expression
mgcoval.1 𝐴 = (Baseβ€˜π‘‰)
mgcoval.2 𝐡 = (Baseβ€˜π‘Š)
mgcoval.3 ≀ = (leβ€˜π‘‰)
mgcoval.4 ≲ = (leβ€˜π‘Š)
Assertion
Ref Expression
mgcoval ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ (𝑉MGalConnπ‘Š) = {βŸ¨π‘“, π‘”βŸ© ∣ ((𝑓 ∈ (𝐡 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐡)) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (π‘”β€˜π‘¦)))})
Distinct variable groups:   𝐴,𝑓,𝑔,π‘₯,𝑦   𝐡,𝑓,𝑔,π‘₯,𝑦   𝑓,𝑉,𝑔,π‘₯,𝑦   𝑓,π‘Š,𝑔,π‘₯,𝑦   𝑓,𝑋,𝑔,π‘₯,𝑦   𝑓,π‘Œ,𝑔,π‘₯,𝑦
Allowed substitution hints:   ≀ (π‘₯,𝑦,𝑓,𝑔)   ≲ (π‘₯,𝑦,𝑓,𝑔)

Proof of Theorem mgcoval
Dummy variables π‘Ž 𝑏 𝑣 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mgc 31890 . . 3 MGalConn = (𝑣 ∈ V, 𝑀 ∈ V ↦ ⦋(Baseβ€˜π‘£) / π‘Žβ¦Œβ¦‹(Baseβ€˜π‘€) / π‘β¦Œ{βŸ¨π‘“, π‘”βŸ© ∣ ((𝑓 ∈ (𝑏 ↑m π‘Ž) ∧ 𝑔 ∈ (π‘Ž ↑m 𝑏)) ∧ βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 ((π‘“β€˜π‘₯)(leβ€˜π‘€)𝑦 ↔ π‘₯(leβ€˜π‘£)(π‘”β€˜π‘¦)))})
21a1i 11 . 2 ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ MGalConn = (𝑣 ∈ V, 𝑀 ∈ V ↦ ⦋(Baseβ€˜π‘£) / π‘Žβ¦Œβ¦‹(Baseβ€˜π‘€) / π‘β¦Œ{βŸ¨π‘“, π‘”βŸ© ∣ ((𝑓 ∈ (𝑏 ↑m π‘Ž) ∧ 𝑔 ∈ (π‘Ž ↑m 𝑏)) ∧ βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 ((π‘“β€˜π‘₯)(leβ€˜π‘€)𝑦 ↔ π‘₯(leβ€˜π‘£)(π‘”β€˜π‘¦)))}))
3 fvexd 6858 . . 3 (((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) β†’ (Baseβ€˜π‘£) ∈ V)
4 simprl 770 . . . . 5 (((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) β†’ 𝑣 = 𝑉)
54fveq2d 6847 . . . 4 (((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) β†’ (Baseβ€˜π‘£) = (Baseβ€˜π‘‰))
6 mgcoval.1 . . . 4 𝐴 = (Baseβ€˜π‘‰)
75, 6eqtr4di 2791 . . 3 (((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) β†’ (Baseβ€˜π‘£) = 𝐴)
8 fvexd 6858 . . . 4 ((((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) ∧ π‘Ž = 𝐴) β†’ (Baseβ€˜π‘€) ∈ V)
9 simplrr 777 . . . . . 6 ((((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) ∧ π‘Ž = 𝐴) β†’ 𝑀 = π‘Š)
109fveq2d 6847 . . . . 5 ((((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) ∧ π‘Ž = 𝐴) β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
11 mgcoval.2 . . . . 5 𝐡 = (Baseβ€˜π‘Š)
1210, 11eqtr4di 2791 . . . 4 ((((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) ∧ π‘Ž = 𝐴) β†’ (Baseβ€˜π‘€) = 𝐡)
13 simpr 486 . . . . . . . . 9 (((((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) ∧ π‘Ž = 𝐴) ∧ 𝑏 = 𝐡) β†’ 𝑏 = 𝐡)
14 simplr 768 . . . . . . . . 9 (((((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) ∧ π‘Ž = 𝐴) ∧ 𝑏 = 𝐡) β†’ π‘Ž = 𝐴)
1513, 14oveq12d 7376 . . . . . . . 8 (((((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) ∧ π‘Ž = 𝐴) ∧ 𝑏 = 𝐡) β†’ (𝑏 ↑m π‘Ž) = (𝐡 ↑m 𝐴))
1615eleq2d 2820 . . . . . . 7 (((((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) ∧ π‘Ž = 𝐴) ∧ 𝑏 = 𝐡) β†’ (𝑓 ∈ (𝑏 ↑m π‘Ž) ↔ 𝑓 ∈ (𝐡 ↑m 𝐴)))
1714, 13oveq12d 7376 . . . . . . . 8 (((((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) ∧ π‘Ž = 𝐴) ∧ 𝑏 = 𝐡) β†’ (π‘Ž ↑m 𝑏) = (𝐴 ↑m 𝐡))
1817eleq2d 2820 . . . . . . 7 (((((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) ∧ π‘Ž = 𝐴) ∧ 𝑏 = 𝐡) β†’ (𝑔 ∈ (π‘Ž ↑m 𝑏) ↔ 𝑔 ∈ (𝐴 ↑m 𝐡)))
1916, 18anbi12d 632 . . . . . 6 (((((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) ∧ π‘Ž = 𝐴) ∧ 𝑏 = 𝐡) β†’ ((𝑓 ∈ (𝑏 ↑m π‘Ž) ∧ 𝑔 ∈ (π‘Ž ↑m 𝑏)) ↔ (𝑓 ∈ (𝐡 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐡))))
209adantr 482 . . . . . . . . . . . 12 (((((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) ∧ π‘Ž = 𝐴) ∧ 𝑏 = 𝐡) β†’ 𝑀 = π‘Š)
2120fveq2d 6847 . . . . . . . . . . 11 (((((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) ∧ π‘Ž = 𝐴) ∧ 𝑏 = 𝐡) β†’ (leβ€˜π‘€) = (leβ€˜π‘Š))
22 mgcoval.4 . . . . . . . . . . 11 ≲ = (leβ€˜π‘Š)
2321, 22eqtr4di 2791 . . . . . . . . . 10 (((((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) ∧ π‘Ž = 𝐴) ∧ 𝑏 = 𝐡) β†’ (leβ€˜π‘€) = ≲ )
2423breqd 5117 . . . . . . . . 9 (((((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) ∧ π‘Ž = 𝐴) ∧ 𝑏 = 𝐡) β†’ ((π‘“β€˜π‘₯)(leβ€˜π‘€)𝑦 ↔ (π‘“β€˜π‘₯) ≲ 𝑦))
254ad2antrr 725 . . . . . . . . . . . 12 (((((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) ∧ π‘Ž = 𝐴) ∧ 𝑏 = 𝐡) β†’ 𝑣 = 𝑉)
2625fveq2d 6847 . . . . . . . . . . 11 (((((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) ∧ π‘Ž = 𝐴) ∧ 𝑏 = 𝐡) β†’ (leβ€˜π‘£) = (leβ€˜π‘‰))
27 mgcoval.3 . . . . . . . . . . 11 ≀ = (leβ€˜π‘‰)
2826, 27eqtr4di 2791 . . . . . . . . . 10 (((((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) ∧ π‘Ž = 𝐴) ∧ 𝑏 = 𝐡) β†’ (leβ€˜π‘£) = ≀ )
2928breqd 5117 . . . . . . . . 9 (((((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) ∧ π‘Ž = 𝐴) ∧ 𝑏 = 𝐡) β†’ (π‘₯(leβ€˜π‘£)(π‘”β€˜π‘¦) ↔ π‘₯ ≀ (π‘”β€˜π‘¦)))
3024, 29bibi12d 346 . . . . . . . 8 (((((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) ∧ π‘Ž = 𝐴) ∧ 𝑏 = 𝐡) β†’ (((π‘“β€˜π‘₯)(leβ€˜π‘€)𝑦 ↔ π‘₯(leβ€˜π‘£)(π‘”β€˜π‘¦)) ↔ ((π‘“β€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (π‘”β€˜π‘¦))))
3113, 30raleqbidv 3318 . . . . . . 7 (((((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) ∧ π‘Ž = 𝐴) ∧ 𝑏 = 𝐡) β†’ (βˆ€π‘¦ ∈ 𝑏 ((π‘“β€˜π‘₯)(leβ€˜π‘€)𝑦 ↔ π‘₯(leβ€˜π‘£)(π‘”β€˜π‘¦)) ↔ βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (π‘”β€˜π‘¦))))
3214, 31raleqbidv 3318 . . . . . 6 (((((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) ∧ π‘Ž = 𝐴) ∧ 𝑏 = 𝐡) β†’ (βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 ((π‘“β€˜π‘₯)(leβ€˜π‘€)𝑦 ↔ π‘₯(leβ€˜π‘£)(π‘”β€˜π‘¦)) ↔ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (π‘”β€˜π‘¦))))
3319, 32anbi12d 632 . . . . 5 (((((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) ∧ π‘Ž = 𝐴) ∧ 𝑏 = 𝐡) β†’ (((𝑓 ∈ (𝑏 ↑m π‘Ž) ∧ 𝑔 ∈ (π‘Ž ↑m 𝑏)) ∧ βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 ((π‘“β€˜π‘₯)(leβ€˜π‘€)𝑦 ↔ π‘₯(leβ€˜π‘£)(π‘”β€˜π‘¦))) ↔ ((𝑓 ∈ (𝐡 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐡)) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (π‘”β€˜π‘¦)))))
3433opabbidv 5172 . . . 4 (((((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) ∧ π‘Ž = 𝐴) ∧ 𝑏 = 𝐡) β†’ {βŸ¨π‘“, π‘”βŸ© ∣ ((𝑓 ∈ (𝑏 ↑m π‘Ž) ∧ 𝑔 ∈ (π‘Ž ↑m 𝑏)) ∧ βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 ((π‘“β€˜π‘₯)(leβ€˜π‘€)𝑦 ↔ π‘₯(leβ€˜π‘£)(π‘”β€˜π‘¦)))} = {βŸ¨π‘“, π‘”βŸ© ∣ ((𝑓 ∈ (𝐡 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐡)) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (π‘”β€˜π‘¦)))})
358, 12, 34csbied2 3896 . . 3 ((((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) ∧ π‘Ž = 𝐴) β†’ ⦋(Baseβ€˜π‘€) / π‘β¦Œ{βŸ¨π‘“, π‘”βŸ© ∣ ((𝑓 ∈ (𝑏 ↑m π‘Ž) ∧ 𝑔 ∈ (π‘Ž ↑m 𝑏)) ∧ βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 ((π‘“β€˜π‘₯)(leβ€˜π‘€)𝑦 ↔ π‘₯(leβ€˜π‘£)(π‘”β€˜π‘¦)))} = {βŸ¨π‘“, π‘”βŸ© ∣ ((𝑓 ∈ (𝐡 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐡)) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (π‘”β€˜π‘¦)))})
363, 7, 35csbied2 3896 . 2 (((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) β†’ ⦋(Baseβ€˜π‘£) / π‘Žβ¦Œβ¦‹(Baseβ€˜π‘€) / π‘β¦Œ{βŸ¨π‘“, π‘”βŸ© ∣ ((𝑓 ∈ (𝑏 ↑m π‘Ž) ∧ 𝑔 ∈ (π‘Ž ↑m 𝑏)) ∧ βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 ((π‘“β€˜π‘₯)(leβ€˜π‘€)𝑦 ↔ π‘₯(leβ€˜π‘£)(π‘”β€˜π‘¦)))} = {βŸ¨π‘“, π‘”βŸ© ∣ ((𝑓 ∈ (𝐡 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐡)) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (π‘”β€˜π‘¦)))})
37 simpl 484 . . 3 ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ 𝑉 ∈ 𝑋)
3837elexd 3464 . 2 ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ 𝑉 ∈ V)
39 simpr 486 . . 3 ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ π‘Š ∈ π‘Œ)
4039elexd 3464 . 2 ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ π‘Š ∈ V)
41 ovexd 7393 . . 3 ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ (𝐡 ↑m 𝐴) ∈ V)
42 ovexd 7393 . . 3 ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ (𝐴 ↑m 𝐡) ∈ V)
43 simprll 778 . . 3 (((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ ((𝑓 ∈ (𝐡 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐡)) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (π‘”β€˜π‘¦)))) β†’ 𝑓 ∈ (𝐡 ↑m 𝐴))
44 simprlr 779 . . 3 (((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ ((𝑓 ∈ (𝐡 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐡)) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (π‘”β€˜π‘¦)))) β†’ 𝑔 ∈ (𝐴 ↑m 𝐡))
4541, 42, 43, 44opabex2 7990 . 2 ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ {βŸ¨π‘“, π‘”βŸ© ∣ ((𝑓 ∈ (𝐡 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐡)) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (π‘”β€˜π‘¦)))} ∈ V)
462, 36, 38, 40, 45ovmpod 7508 1 ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ (𝑉MGalConnπ‘Š) = {βŸ¨π‘“, π‘”βŸ© ∣ ((𝑓 ∈ (𝐡 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐡)) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (π‘”β€˜π‘¦)))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  Vcvv 3444  β¦‹csb 3856   class class class wbr 5106  {copab 5168  β€˜cfv 6497  (class class class)co 7358   ∈ cmpo 7360   ↑m cmap 8768  Basecbs 17088  lecple 17145  MGalConncmgc 31888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-mgc 31890
This theorem is referenced by:  mgcval  31896
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