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Theorem legval 28102
Description: Value of the less-than relationship. (Contributed by Thierry Arnoux, 21-Jun-2019.)
Hypotheses
Ref Expression
legval.p 𝑃 = (Baseβ€˜πΊ)
legval.d βˆ’ = (distβ€˜πΊ)
legval.i 𝐼 = (Itvβ€˜πΊ)
legval.l ≀ = (≀Gβ€˜πΊ)
legval.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
Assertion
Ref Expression
legval (πœ‘ β†’ ≀ = {βŸ¨π‘’, π‘“βŸ© ∣ βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 (𝑓 = (π‘₯ βˆ’ 𝑦) ∧ βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧)))})
Distinct variable groups:   𝑒,𝑓,𝐺   π‘₯,𝑦,𝑧,𝐼   π‘₯,𝑒,𝑦,𝑧,𝑃,𝑓   βˆ’ ,𝑒,𝑓,π‘₯,𝑦,𝑧
Allowed substitution hints:   πœ‘(π‘₯,𝑦,𝑧,𝑒,𝑓)   𝐺(π‘₯,𝑦,𝑧)   𝐼(𝑒,𝑓)   ≀ (π‘₯,𝑦,𝑧,𝑒,𝑓)

Proof of Theorem legval
Dummy variables 𝑑 𝑔 𝑖 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 legval.l . 2 ≀ = (≀Gβ€˜πΊ)
2 legval.g . . 3 (πœ‘ β†’ 𝐺 ∈ TarskiG)
3 elex 3491 . . 3 (𝐺 ∈ TarskiG β†’ 𝐺 ∈ V)
4 legval.p . . . . . 6 𝑃 = (Baseβ€˜πΊ)
5 legval.d . . . . . 6 βˆ’ = (distβ€˜πΊ)
6 legval.i . . . . . 6 𝐼 = (Itvβ€˜πΊ)
7 simp1 1134 . . . . . . . 8 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ∧ 𝑖 = 𝐼) β†’ 𝑝 = 𝑃)
87eqcomd 2736 . . . . . . 7 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ∧ 𝑖 = 𝐼) β†’ 𝑃 = 𝑝)
9 simp2 1135 . . . . . . . . . . . 12 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ∧ 𝑖 = 𝐼) β†’ 𝑑 = βˆ’ )
109eqcomd 2736 . . . . . . . . . . 11 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ∧ 𝑖 = 𝐼) β†’ βˆ’ = 𝑑)
1110oveqd 7428 . . . . . . . . . 10 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ∧ 𝑖 = 𝐼) β†’ (π‘₯ βˆ’ 𝑦) = (π‘₯𝑑𝑦))
1211eqeq2d 2741 . . . . . . . . 9 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ∧ 𝑖 = 𝐼) β†’ (𝑓 = (π‘₯ βˆ’ 𝑦) ↔ 𝑓 = (π‘₯𝑑𝑦)))
13 simp3 1136 . . . . . . . . . . . . . 14 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ∧ 𝑖 = 𝐼) β†’ 𝑖 = 𝐼)
1413eqcomd 2736 . . . . . . . . . . . . 13 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ∧ 𝑖 = 𝐼) β†’ 𝐼 = 𝑖)
1514oveqd 7428 . . . . . . . . . . . 12 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ∧ 𝑖 = 𝐼) β†’ (π‘₯𝐼𝑦) = (π‘₯𝑖𝑦))
1615eleq2d 2817 . . . . . . . . . . 11 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ∧ 𝑖 = 𝐼) β†’ (𝑧 ∈ (π‘₯𝐼𝑦) ↔ 𝑧 ∈ (π‘₯𝑖𝑦)))
1710oveqd 7428 . . . . . . . . . . . 12 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ∧ 𝑖 = 𝐼) β†’ (π‘₯ βˆ’ 𝑧) = (π‘₯𝑑𝑧))
1817eqeq2d 2741 . . . . . . . . . . 11 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ∧ 𝑖 = 𝐼) β†’ (𝑒 = (π‘₯ βˆ’ 𝑧) ↔ 𝑒 = (π‘₯𝑑𝑧)))
1916, 18anbi12d 629 . . . . . . . . . 10 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ∧ 𝑖 = 𝐼) β†’ ((𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧)) ↔ (𝑧 ∈ (π‘₯𝑖𝑦) ∧ 𝑒 = (π‘₯𝑑𝑧))))
208, 19rexeqbidv 3341 . . . . . . . . 9 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ∧ 𝑖 = 𝐼) β†’ (βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧)) ↔ βˆƒπ‘§ ∈ 𝑝 (𝑧 ∈ (π‘₯𝑖𝑦) ∧ 𝑒 = (π‘₯𝑑𝑧))))
2112, 20anbi12d 629 . . . . . . . 8 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ∧ 𝑖 = 𝐼) β†’ ((𝑓 = (π‘₯ βˆ’ 𝑦) ∧ βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧))) ↔ (𝑓 = (π‘₯𝑑𝑦) ∧ βˆƒπ‘§ ∈ 𝑝 (𝑧 ∈ (π‘₯𝑖𝑦) ∧ 𝑒 = (π‘₯𝑑𝑧)))))
228, 21rexeqbidv 3341 . . . . . . 7 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ∧ 𝑖 = 𝐼) β†’ (βˆƒπ‘¦ ∈ 𝑃 (𝑓 = (π‘₯ βˆ’ 𝑦) ∧ βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧))) ↔ βˆƒπ‘¦ ∈ 𝑝 (𝑓 = (π‘₯𝑑𝑦) ∧ βˆƒπ‘§ ∈ 𝑝 (𝑧 ∈ (π‘₯𝑖𝑦) ∧ 𝑒 = (π‘₯𝑑𝑧)))))
238, 22rexeqbidv 3341 . . . . . 6 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ∧ 𝑖 = 𝐼) β†’ (βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 (𝑓 = (π‘₯ βˆ’ 𝑦) ∧ βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧))) ↔ βˆƒπ‘₯ ∈ 𝑝 βˆƒπ‘¦ ∈ 𝑝 (𝑓 = (π‘₯𝑑𝑦) ∧ βˆƒπ‘§ ∈ 𝑝 (𝑧 ∈ (π‘₯𝑖𝑦) ∧ 𝑒 = (π‘₯𝑑𝑧)))))
244, 5, 6, 23sbcie3s 17099 . . . . 5 (𝑔 = 𝐺 β†’ ([(Baseβ€˜π‘”) / 𝑝][(distβ€˜π‘”) / 𝑑][(Itvβ€˜π‘”) / 𝑖]βˆƒπ‘₯ ∈ 𝑝 βˆƒπ‘¦ ∈ 𝑝 (𝑓 = (π‘₯𝑑𝑦) ∧ βˆƒπ‘§ ∈ 𝑝 (𝑧 ∈ (π‘₯𝑖𝑦) ∧ 𝑒 = (π‘₯𝑑𝑧))) ↔ βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 (𝑓 = (π‘₯ βˆ’ 𝑦) ∧ βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧)))))
2524opabbidv 5213 . . . 4 (𝑔 = 𝐺 β†’ {βŸ¨π‘’, π‘“βŸ© ∣ [(Baseβ€˜π‘”) / 𝑝][(distβ€˜π‘”) / 𝑑][(Itvβ€˜π‘”) / 𝑖]βˆƒπ‘₯ ∈ 𝑝 βˆƒπ‘¦ ∈ 𝑝 (𝑓 = (π‘₯𝑑𝑦) ∧ βˆƒπ‘§ ∈ 𝑝 (𝑧 ∈ (π‘₯𝑖𝑦) ∧ 𝑒 = (π‘₯𝑑𝑧)))} = {βŸ¨π‘’, π‘“βŸ© ∣ βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 (𝑓 = (π‘₯ βˆ’ 𝑦) ∧ βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧)))})
26 df-leg 28101 . . . 4 ≀G = (𝑔 ∈ V ↦ {βŸ¨π‘’, π‘“βŸ© ∣ [(Baseβ€˜π‘”) / 𝑝][(distβ€˜π‘”) / 𝑑][(Itvβ€˜π‘”) / 𝑖]βˆƒπ‘₯ ∈ 𝑝 βˆƒπ‘¦ ∈ 𝑝 (𝑓 = (π‘₯𝑑𝑦) ∧ βˆƒπ‘§ ∈ 𝑝 (𝑧 ∈ (π‘₯𝑖𝑦) ∧ 𝑒 = (π‘₯𝑑𝑧)))})
275fvexi 6904 . . . . . . . . 9 βˆ’ ∈ V
2827imaex 7909 . . . . . . . 8 ( βˆ’ β€œ (𝑃 Γ— 𝑃)) ∈ V
29 p0ex 5381 . . . . . . . 8 {βˆ…} ∈ V
3028, 29unex 7735 . . . . . . 7 (( βˆ’ β€œ (𝑃 Γ— 𝑃)) βˆͺ {βˆ…}) ∈ V
3130a1i 11 . . . . . 6 (⊀ β†’ (( βˆ’ β€œ (𝑃 Γ— 𝑃)) βˆͺ {βˆ…}) ∈ V)
32 simprr 769 . . . . . . . . . . . . 13 (((((π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ (𝑓 = (π‘₯ βˆ’ 𝑦) ∧ βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧)))) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑑))) β†’ 𝑒 = (π‘₯ βˆ’ 𝑑))
33 ovima0 7588 . . . . . . . . . . . . . 14 ((π‘₯ ∈ 𝑃 ∧ 𝑑 ∈ 𝑃) β†’ (π‘₯ βˆ’ 𝑑) ∈ (( βˆ’ β€œ (𝑃 Γ— 𝑃)) βˆͺ {βˆ…}))
3433ad5ant14 754 . . . . . . . . . . . . 13 (((((π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ (𝑓 = (π‘₯ βˆ’ 𝑦) ∧ βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧)))) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑑))) β†’ (π‘₯ βˆ’ 𝑑) ∈ (( βˆ’ β€œ (𝑃 Γ— 𝑃)) βˆͺ {βˆ…}))
3532, 34eqeltrd 2831 . . . . . . . . . . . 12 (((((π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ (𝑓 = (π‘₯ βˆ’ 𝑦) ∧ βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧)))) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑑))) β†’ 𝑒 ∈ (( βˆ’ β€œ (𝑃 Γ— 𝑃)) βˆͺ {βˆ…}))
36 simpllr 772 . . . . . . . . . . . . . 14 (((((π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ (𝑓 = (π‘₯ βˆ’ 𝑦) ∧ βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧)))) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑑))) β†’ (𝑓 = (π‘₯ βˆ’ 𝑦) ∧ βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧))))
3736simpld 493 . . . . . . . . . . . . 13 (((((π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ (𝑓 = (π‘₯ βˆ’ 𝑦) ∧ βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧)))) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑑))) β†’ 𝑓 = (π‘₯ βˆ’ 𝑦))
38 ovima0 7588 . . . . . . . . . . . . . 14 ((π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) β†’ (π‘₯ βˆ’ 𝑦) ∈ (( βˆ’ β€œ (𝑃 Γ— 𝑃)) βˆͺ {βˆ…}))
3938ad3antrrr 726 . . . . . . . . . . . . 13 (((((π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ (𝑓 = (π‘₯ βˆ’ 𝑦) ∧ βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧)))) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑑))) β†’ (π‘₯ βˆ’ 𝑦) ∈ (( βˆ’ β€œ (𝑃 Γ— 𝑃)) βˆͺ {βˆ…}))
4037, 39eqeltrd 2831 . . . . . . . . . . . 12 (((((π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ (𝑓 = (π‘₯ βˆ’ 𝑦) ∧ βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧)))) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑑))) β†’ 𝑓 ∈ (( βˆ’ β€œ (𝑃 Γ— 𝑃)) βˆͺ {βˆ…}))
4135, 40jca 510 . . . . . . . . . . 11 (((((π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ (𝑓 = (π‘₯ βˆ’ 𝑦) ∧ βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧)))) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑑))) β†’ (𝑒 ∈ (( βˆ’ β€œ (𝑃 Γ— 𝑃)) βˆͺ {βˆ…}) ∧ 𝑓 ∈ (( βˆ’ β€œ (𝑃 Γ— 𝑃)) βˆͺ {βˆ…})))
42 simprr 769 . . . . . . . . . . . 12 (((π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ (𝑓 = (π‘₯ βˆ’ 𝑦) ∧ βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧)))) β†’ βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧)))
43 eleq1w 2814 . . . . . . . . . . . . . 14 (𝑧 = 𝑑 β†’ (𝑧 ∈ (π‘₯𝐼𝑦) ↔ 𝑑 ∈ (π‘₯𝐼𝑦)))
44 oveq2 7419 . . . . . . . . . . . . . . 15 (𝑧 = 𝑑 β†’ (π‘₯ βˆ’ 𝑧) = (π‘₯ βˆ’ 𝑑))
4544eqeq2d 2741 . . . . . . . . . . . . . 14 (𝑧 = 𝑑 β†’ (𝑒 = (π‘₯ βˆ’ 𝑧) ↔ 𝑒 = (π‘₯ βˆ’ 𝑑)))
4643, 45anbi12d 629 . . . . . . . . . . . . 13 (𝑧 = 𝑑 β†’ ((𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧)) ↔ (𝑑 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑑))))
4746cbvrexvw 3233 . . . . . . . . . . . 12 (βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧)) ↔ βˆƒπ‘‘ ∈ 𝑃 (𝑑 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑑)))
4842, 47sylib 217 . . . . . . . . . . 11 (((π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ (𝑓 = (π‘₯ βˆ’ 𝑦) ∧ βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧)))) β†’ βˆƒπ‘‘ ∈ 𝑃 (𝑑 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑑)))
4941, 48r19.29a 3160 . . . . . . . . . 10 (((π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ (𝑓 = (π‘₯ βˆ’ 𝑦) ∧ βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧)))) β†’ (𝑒 ∈ (( βˆ’ β€œ (𝑃 Γ— 𝑃)) βˆͺ {βˆ…}) ∧ 𝑓 ∈ (( βˆ’ β€œ (𝑃 Γ— 𝑃)) βˆͺ {βˆ…})))
5049ex 411 . . . . . . . . 9 ((π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) β†’ ((𝑓 = (π‘₯ βˆ’ 𝑦) ∧ βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧))) β†’ (𝑒 ∈ (( βˆ’ β€œ (𝑃 Γ— 𝑃)) βˆͺ {βˆ…}) ∧ 𝑓 ∈ (( βˆ’ β€œ (𝑃 Γ— 𝑃)) βˆͺ {βˆ…}))))
5150rexlimivv 3197 . . . . . . . 8 (βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 (𝑓 = (π‘₯ βˆ’ 𝑦) ∧ βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧))) β†’ (𝑒 ∈ (( βˆ’ β€œ (𝑃 Γ— 𝑃)) βˆͺ {βˆ…}) ∧ 𝑓 ∈ (( βˆ’ β€œ (𝑃 Γ— 𝑃)) βˆͺ {βˆ…})))
5251adantl 480 . . . . . . 7 ((⊀ ∧ βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 (𝑓 = (π‘₯ βˆ’ 𝑦) ∧ βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧)))) β†’ (𝑒 ∈ (( βˆ’ β€œ (𝑃 Γ— 𝑃)) βˆͺ {βˆ…}) ∧ 𝑓 ∈ (( βˆ’ β€œ (𝑃 Γ— 𝑃)) βˆͺ {βˆ…})))
5352simpld 493 . . . . . 6 ((⊀ ∧ βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 (𝑓 = (π‘₯ βˆ’ 𝑦) ∧ βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧)))) β†’ 𝑒 ∈ (( βˆ’ β€œ (𝑃 Γ— 𝑃)) βˆͺ {βˆ…}))
5452simprd 494 . . . . . 6 ((⊀ ∧ βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 (𝑓 = (π‘₯ βˆ’ 𝑦) ∧ βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧)))) β†’ 𝑓 ∈ (( βˆ’ β€œ (𝑃 Γ— 𝑃)) βˆͺ {βˆ…}))
5531, 31, 53, 54opabex2 8045 . . . . 5 (⊀ β†’ {βŸ¨π‘’, π‘“βŸ© ∣ βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 (𝑓 = (π‘₯ βˆ’ 𝑦) ∧ βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧)))} ∈ V)
5655mptru 1546 . . . 4 {βŸ¨π‘’, π‘“βŸ© ∣ βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 (𝑓 = (π‘₯ βˆ’ 𝑦) ∧ βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧)))} ∈ V
5725, 26, 56fvmpt 6997 . . 3 (𝐺 ∈ V β†’ (≀Gβ€˜πΊ) = {βŸ¨π‘’, π‘“βŸ© ∣ βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 (𝑓 = (π‘₯ βˆ’ 𝑦) ∧ βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧)))})
582, 3, 573syl 18 . 2 (πœ‘ β†’ (≀Gβ€˜πΊ) = {βŸ¨π‘’, π‘“βŸ© ∣ βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 (𝑓 = (π‘₯ βˆ’ 𝑦) ∧ βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧)))})
591, 58eqtrid 2782 1 (πœ‘ β†’ ≀ = {βŸ¨π‘’, π‘“βŸ© ∣ βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 (𝑓 = (π‘₯ βˆ’ 𝑦) ∧ βˆƒπ‘§ ∈ 𝑃 (𝑧 ∈ (π‘₯𝐼𝑦) ∧ 𝑒 = (π‘₯ βˆ’ 𝑧)))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539  βŠ€wtru 1540   ∈ wcel 2104  βˆƒwrex 3068  Vcvv 3472  [wsbc 3776   βˆͺ cun 3945  βˆ…c0 4321  {csn 4627  {copab 5209   Γ— cxp 5673   β€œ cima 5678  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148  distcds 17210  TarskiGcstrkg 27945  Itvcitv 27951  β‰€Gcleg 28100
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7414  df-leg 28101
This theorem is referenced by:  legov  28103
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