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Mirrors  >  Home  >  MPE Home  >  Th. List  >  injresinjlem Structured version   Visualization version   GIF version
Theorem injresinjlem 13754
Description: Lemma for injresinj 13755. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Proof shortened by AV, 14-Feb-2021.) (Revised by Thierry Arnoux, 23-Dec-2021.)
Assertion
Ref Expression
injresinjlem (Β¬ π‘Œ ∈ (1..^𝐾) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (((𝐹 β€œ {0, 𝐾}) ∩ (𝐹 β€œ (1..^𝐾))) = βˆ… β†’ ((𝑋 ∈ (0...𝐾) ∧ π‘Œ ∈ (0...𝐾)) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ))))))
Proof of Theorem injresinjlem
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 elfznelfzo 13739 . . . . . . 7 ((π‘Œ ∈ (0...𝐾) ∧ Β¬ π‘Œ ∈ (1..^𝐾)) β†’ (π‘Œ = 0 ∨ π‘Œ = 𝐾))
2 fvinim0ffz 13753 . . . . . . . . . . . . 13 ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (((𝐹 β€œ {0, 𝐾}) ∩ (𝐹 β€œ (1..^𝐾))) = βˆ… ↔ ((πΉβ€˜0) βˆ‰ (𝐹 β€œ (1..^𝐾)) ∧ (πΉβ€˜πΎ) βˆ‰ (𝐹 β€œ (1..^𝐾)))))
3 df-nel 3047 . . . . . . . . . . . . . . . . . 18 ((πΉβ€˜0) βˆ‰ (𝐹 β€œ (1..^𝐾)) ↔ Β¬ (πΉβ€˜0) ∈ (𝐹 β€œ (1..^𝐾)))
4 fveq2 6891 . . . . . . . . . . . . . . . . . . . . . . 23 (0 = π‘Œ β†’ (πΉβ€˜0) = (πΉβ€˜π‘Œ))
54eqcoms 2740 . . . . . . . . . . . . . . . . . . . . . 22 (π‘Œ = 0 β†’ (πΉβ€˜0) = (πΉβ€˜π‘Œ))
65eleq1d 2818 . . . . . . . . . . . . . . . . . . . . 21 (π‘Œ = 0 β†’ ((πΉβ€˜0) ∈ (𝐹 β€œ (1..^𝐾)) ↔ (πΉβ€˜π‘Œ) ∈ (𝐹 β€œ (1..^𝐾))))
76notbid 317 . . . . . . . . . . . . . . . . . . . 20 (π‘Œ = 0 β†’ (Β¬ (πΉβ€˜0) ∈ (𝐹 β€œ (1..^𝐾)) ↔ Β¬ (πΉβ€˜π‘Œ) ∈ (𝐹 β€œ (1..^𝐾))))
87biimpd 228 . . . . . . . . . . . . . . . . . . 19 (π‘Œ = 0 β†’ (Β¬ (πΉβ€˜0) ∈ (𝐹 β€œ (1..^𝐾)) β†’ Β¬ (πΉβ€˜π‘Œ) ∈ (𝐹 β€œ (1..^𝐾))))
9 ffn 6717 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:(0...𝐾)βŸΆπ‘‰ β†’ 𝐹 Fn (0...𝐾))
10 1eluzge0 12878 . . . . . . . . . . . . . . . . . . . . . . . . 25 1 ∈ (β„€β‰₯β€˜0)
11 fzoss1 13661 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1 ∈ (β„€β‰₯β€˜0) β†’ (1..^𝐾) βŠ† (0..^𝐾))
1210, 11mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐾 ∈ β„•0 β†’ (1..^𝐾) βŠ† (0..^𝐾))
13 fzossfz 13653 . . . . . . . . . . . . . . . . . . . . . . . 24 (0..^𝐾) βŠ† (0...𝐾)
1412, 13sstrdi 3994 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐾 ∈ β„•0 β†’ (1..^𝐾) βŠ† (0...𝐾))
15 fvelimab 6964 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹 Fn (0...𝐾) ∧ (1..^𝐾) βŠ† (0...𝐾)) β†’ ((πΉβ€˜π‘Œ) ∈ (𝐹 β€œ (1..^𝐾)) ↔ βˆƒπ‘§ ∈ (1..^𝐾)(πΉβ€˜π‘§) = (πΉβ€˜π‘Œ)))
169, 14, 15syl2an 596 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ ((πΉβ€˜π‘Œ) ∈ (𝐹 β€œ (1..^𝐾)) ↔ βˆƒπ‘§ ∈ (1..^𝐾)(πΉβ€˜π‘§) = (πΉβ€˜π‘Œ)))
1716notbid 317 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (Β¬ (πΉβ€˜π‘Œ) ∈ (𝐹 β€œ (1..^𝐾)) ↔ Β¬ βˆƒπ‘§ ∈ (1..^𝐾)(πΉβ€˜π‘§) = (πΉβ€˜π‘Œ)))
18 ralnex 3072 . . . . . . . . . . . . . . . . . . . . . . 23 (βˆ€π‘§ ∈ (1..^𝐾) Β¬ (πΉβ€˜π‘§) = (πΉβ€˜π‘Œ) ↔ Β¬ βˆƒπ‘§ ∈ (1..^𝐾)(πΉβ€˜π‘§) = (πΉβ€˜π‘Œ))
19 fveqeq2 6900 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = 𝑋 β†’ ((πΉβ€˜π‘§) = (πΉβ€˜π‘Œ) ↔ (πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ)))
2019notbid 317 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 𝑋 β†’ (Β¬ (πΉβ€˜π‘§) = (πΉβ€˜π‘Œ) ↔ Β¬ (πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ)))
2120rspcva 3610 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑋 ∈ (1..^𝐾) ∧ βˆ€π‘§ ∈ (1..^𝐾) Β¬ (πΉβ€˜π‘§) = (πΉβ€˜π‘Œ)) β†’ Β¬ (πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ))
22 pm2.21 123 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Β¬ (πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ))
2322a1d 25 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Β¬ (πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ)))
24232a1d 26 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (Β¬ (πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ (𝑋 ∈ (0...𝐾) β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ)))))
2521, 24syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑋 ∈ (1..^𝐾) ∧ βˆ€π‘§ ∈ (1..^𝐾) Β¬ (πΉβ€˜π‘§) = (πΉβ€˜π‘Œ)) β†’ (𝑋 ∈ (0...𝐾) β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ)))))
2625expcom 414 . . . . . . . . . . . . . . . . . . . . . . . 24 (βˆ€π‘§ ∈ (1..^𝐾) Β¬ (πΉβ€˜π‘§) = (πΉβ€˜π‘Œ) β†’ (𝑋 ∈ (1..^𝐾) β†’ (𝑋 ∈ (0...𝐾) β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ))))))
2726com24 95 . . . . . . . . . . . . . . . . . . . . . . 23 (βˆ€π‘§ ∈ (1..^𝐾) Β¬ (πΉβ€˜π‘§) = (πΉβ€˜π‘Œ) β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (𝑋 ∈ (0...𝐾) β†’ (𝑋 ∈ (1..^𝐾) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ))))))
2818, 27sylbir 234 . . . . . . . . . . . . . . . . . . . . . 22 (Β¬ βˆƒπ‘§ ∈ (1..^𝐾)(πΉβ€˜π‘§) = (πΉβ€˜π‘Œ) β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (𝑋 ∈ (0...𝐾) β†’ (𝑋 ∈ (1..^𝐾) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ))))))
2928com12 32 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (Β¬ βˆƒπ‘§ ∈ (1..^𝐾)(πΉβ€˜π‘§) = (πΉβ€˜π‘Œ) β†’ (𝑋 ∈ (0...𝐾) β†’ (𝑋 ∈ (1..^𝐾) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ))))))
3017, 29sylbid 239 . . . . . . . . . . . . . . . . . . . 20 ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (Β¬ (πΉβ€˜π‘Œ) ∈ (𝐹 β€œ (1..^𝐾)) β†’ (𝑋 ∈ (0...𝐾) β†’ (𝑋 ∈ (1..^𝐾) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ))))))
3130com12 32 . . . . . . . . . . . . . . . . . . 19 (Β¬ (πΉβ€˜π‘Œ) ∈ (𝐹 β€œ (1..^𝐾)) β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (𝑋 ∈ (0...𝐾) β†’ (𝑋 ∈ (1..^𝐾) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ))))))
328, 31syl6com 37 . . . . . . . . . . . . . . . . . 18 (Β¬ (πΉβ€˜0) ∈ (𝐹 β€œ (1..^𝐾)) β†’ (π‘Œ = 0 β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (𝑋 ∈ (0...𝐾) β†’ (𝑋 ∈ (1..^𝐾) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ)))))))
333, 32sylbi 216 . . . . . . . . . . . . . . . . 17 ((πΉβ€˜0) βˆ‰ (𝐹 β€œ (1..^𝐾)) β†’ (π‘Œ = 0 β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (𝑋 ∈ (0...𝐾) β†’ (𝑋 ∈ (1..^𝐾) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ)))))))
3433adantr 481 . . . . . . . . . . . . . . . 16 (((πΉβ€˜0) βˆ‰ (𝐹 β€œ (1..^𝐾)) ∧ (πΉβ€˜πΎ) βˆ‰ (𝐹 β€œ (1..^𝐾))) β†’ (π‘Œ = 0 β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (𝑋 ∈ (0...𝐾) β†’ (𝑋 ∈ (1..^𝐾) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ)))))))
3534com12 32 . . . . . . . . . . . . . . 15 (π‘Œ = 0 β†’ (((πΉβ€˜0) βˆ‰ (𝐹 β€œ (1..^𝐾)) ∧ (πΉβ€˜πΎ) βˆ‰ (𝐹 β€œ (1..^𝐾))) β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (𝑋 ∈ (0...𝐾) β†’ (𝑋 ∈ (1..^𝐾) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ)))))))
36 df-nel 3047 . . . . . . . . . . . . . . . . . 18 ((πΉβ€˜πΎ) βˆ‰ (𝐹 β€œ (1..^𝐾)) ↔ Β¬ (πΉβ€˜πΎ) ∈ (𝐹 β€œ (1..^𝐾)))
37 fveq2 6891 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐾 = π‘Œ β†’ (πΉβ€˜πΎ) = (πΉβ€˜π‘Œ))
3837eqcoms 2740 . . . . . . . . . . . . . . . . . . . . . 22 (π‘Œ = 𝐾 β†’ (πΉβ€˜πΎ) = (πΉβ€˜π‘Œ))
3938eleq1d 2818 . . . . . . . . . . . . . . . . . . . . 21 (π‘Œ = 𝐾 β†’ ((πΉβ€˜πΎ) ∈ (𝐹 β€œ (1..^𝐾)) ↔ (πΉβ€˜π‘Œ) ∈ (𝐹 β€œ (1..^𝐾))))
4039notbid 317 . . . . . . . . . . . . . . . . . . . 20 (π‘Œ = 𝐾 β†’ (Β¬ (πΉβ€˜πΎ) ∈ (𝐹 β€œ (1..^𝐾)) ↔ Β¬ (πΉβ€˜π‘Œ) ∈ (𝐹 β€œ (1..^𝐾))))
4140biimpd 228 . . . . . . . . . . . . . . . . . . 19 (π‘Œ = 𝐾 β†’ (Β¬ (πΉβ€˜πΎ) ∈ (𝐹 β€œ (1..^𝐾)) β†’ Β¬ (πΉβ€˜π‘Œ) ∈ (𝐹 β€œ (1..^𝐾))))
4241, 31syl6com 37 . . . . . . . . . . . . . . . . . 18 (Β¬ (πΉβ€˜πΎ) ∈ (𝐹 β€œ (1..^𝐾)) β†’ (π‘Œ = 𝐾 β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (𝑋 ∈ (0...𝐾) β†’ (𝑋 ∈ (1..^𝐾) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ)))))))
4336, 42sylbi 216 . . . . . . . . . . . . . . . . 17 ((πΉβ€˜πΎ) βˆ‰ (𝐹 β€œ (1..^𝐾)) β†’ (π‘Œ = 𝐾 β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (𝑋 ∈ (0...𝐾) β†’ (𝑋 ∈ (1..^𝐾) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ)))))))
4443adantl 482 . . . . . . . . . . . . . . . 16 (((πΉβ€˜0) βˆ‰ (𝐹 β€œ (1..^𝐾)) ∧ (πΉβ€˜πΎ) βˆ‰ (𝐹 β€œ (1..^𝐾))) β†’ (π‘Œ = 𝐾 β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (𝑋 ∈ (0...𝐾) β†’ (𝑋 ∈ (1..^𝐾) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ)))))))
4544com12 32 . . . . . . . . . . . . . . 15 (π‘Œ = 𝐾 β†’ (((πΉβ€˜0) βˆ‰ (𝐹 β€œ (1..^𝐾)) ∧ (πΉβ€˜πΎ) βˆ‰ (𝐹 β€œ (1..^𝐾))) β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (𝑋 ∈ (0...𝐾) β†’ (𝑋 ∈ (1..^𝐾) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ)))))))
4635, 45jaoi 855 . . . . . . . . . . . . . 14 ((π‘Œ = 0 ∨ π‘Œ = 𝐾) β†’ (((πΉβ€˜0) βˆ‰ (𝐹 β€œ (1..^𝐾)) ∧ (πΉβ€˜πΎ) βˆ‰ (𝐹 β€œ (1..^𝐾))) β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (𝑋 ∈ (0...𝐾) β†’ (𝑋 ∈ (1..^𝐾) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ)))))))
4746com13 88 . . . . . . . . . . . . 13 ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (((πΉβ€˜0) βˆ‰ (𝐹 β€œ (1..^𝐾)) ∧ (πΉβ€˜πΎ) βˆ‰ (𝐹 β€œ (1..^𝐾))) β†’ ((π‘Œ = 0 ∨ π‘Œ = 𝐾) β†’ (𝑋 ∈ (0...𝐾) β†’ (𝑋 ∈ (1..^𝐾) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ)))))))
482, 47sylbid 239 . . . . . . . . . . . 12 ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (((𝐹 β€œ {0, 𝐾}) ∩ (𝐹 β€œ (1..^𝐾))) = βˆ… β†’ ((π‘Œ = 0 ∨ π‘Œ = 𝐾) β†’ (𝑋 ∈ (0...𝐾) β†’ (𝑋 ∈ (1..^𝐾) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ)))))))
4948com14 96 . . . . . . . . . . 11 (𝑋 ∈ (0...𝐾) β†’ (((𝐹 β€œ {0, 𝐾}) ∩ (𝐹 β€œ (1..^𝐾))) = βˆ… β†’ ((π‘Œ = 0 ∨ π‘Œ = 𝐾) β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (𝑋 ∈ (1..^𝐾) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ)))))))
5049com12 32 . . . . . . . . . 10 (((𝐹 β€œ {0, 𝐾}) ∩ (𝐹 β€œ (1..^𝐾))) = βˆ… β†’ (𝑋 ∈ (0...𝐾) β†’ ((π‘Œ = 0 ∨ π‘Œ = 𝐾) β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (𝑋 ∈ (1..^𝐾) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ)))))))
5150com15 101 . . . . . . . . 9 (𝑋 ∈ (1..^𝐾) β†’ (𝑋 ∈ (0...𝐾) β†’ ((π‘Œ = 0 ∨ π‘Œ = 𝐾) β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (((𝐹 β€œ {0, 𝐾}) ∩ (𝐹 β€œ (1..^𝐾))) = βˆ… β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ)))))))
52 elfznelfzo 13739 . . . . . . . . . . 11 ((𝑋 ∈ (0...𝐾) ∧ Β¬ 𝑋 ∈ (1..^𝐾)) β†’ (𝑋 = 0 ∨ 𝑋 = 𝐾))
53 eqtr3 2758 . . . . . . . . . . . . . 14 ((𝑋 = 0 ∧ π‘Œ = 0) β†’ 𝑋 = π‘Œ)
54 2a1 28 . . . . . . . . . . . . . . 15 (𝑋 = π‘Œ β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ)))
55542a1d 26 . . . . . . . . . . . . . 14 (𝑋 = π‘Œ β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (((𝐹 β€œ {0, 𝐾}) ∩ (𝐹 β€œ (1..^𝐾))) = βˆ… β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ)))))
5653, 55syl 17 . . . . . . . . . . . . 13 ((𝑋 = 0 ∧ π‘Œ = 0) β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (((𝐹 β€œ {0, 𝐾}) ∩ (𝐹 β€œ (1..^𝐾))) = βˆ… β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ)))))
575adantl 482 . . . . . . . . . . . . . . . 16 ((𝑋 = 𝐾 ∧ π‘Œ = 0) β†’ (πΉβ€˜0) = (πΉβ€˜π‘Œ))
58 fveq2 6891 . . . . . . . . . . . . . . . . . 18 (𝐾 = 𝑋 β†’ (πΉβ€˜πΎ) = (πΉβ€˜π‘‹))
5958eqcoms 2740 . . . . . . . . . . . . . . . . 17 (𝑋 = 𝐾 β†’ (πΉβ€˜πΎ) = (πΉβ€˜π‘‹))
6059adantr 481 . . . . . . . . . . . . . . . 16 ((𝑋 = 𝐾 ∧ π‘Œ = 0) β†’ (πΉβ€˜πΎ) = (πΉβ€˜π‘‹))
6157, 60neeq12d 3002 . . . . . . . . . . . . . . 15 ((𝑋 = 𝐾 ∧ π‘Œ = 0) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) ↔ (πΉβ€˜π‘Œ) β‰  (πΉβ€˜π‘‹)))
62 df-ne 2941 . . . . . . . . . . . . . . . 16 ((πΉβ€˜π‘Œ) β‰  (πΉβ€˜π‘‹) ↔ Β¬ (πΉβ€˜π‘Œ) = (πΉβ€˜π‘‹))
63 pm2.24 124 . . . . . . . . . . . . . . . . . 18 ((πΉβ€˜π‘Œ) = (πΉβ€˜π‘‹) β†’ (Β¬ (πΉβ€˜π‘Œ) = (πΉβ€˜π‘‹) β†’ 𝑋 = π‘Œ))
6463eqcoms 2740 . . . . . . . . . . . . . . . . 17 ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ (Β¬ (πΉβ€˜π‘Œ) = (πΉβ€˜π‘‹) β†’ 𝑋 = π‘Œ))
6564com12 32 . . . . . . . . . . . . . . . 16 (Β¬ (πΉβ€˜π‘Œ) = (πΉβ€˜π‘‹) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ))
6662, 65sylbi 216 . . . . . . . . . . . . . . 15 ((πΉβ€˜π‘Œ) β‰  (πΉβ€˜π‘‹) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ))
6761, 66syl6bi 252 . . . . . . . . . . . . . 14 ((𝑋 = 𝐾 ∧ π‘Œ = 0) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ)))
68672a1d 26 . . . . . . . . . . . . 13 ((𝑋 = 𝐾 ∧ π‘Œ = 0) β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (((𝐹 β€œ {0, 𝐾}) ∩ (𝐹 β€œ (1..^𝐾))) = βˆ… β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ)))))
69 fveq2 6891 . . . . . . . . . . . . . . . . . 18 (0 = 𝑋 β†’ (πΉβ€˜0) = (πΉβ€˜π‘‹))
7069eqcoms 2740 . . . . . . . . . . . . . . . . 17 (𝑋 = 0 β†’ (πΉβ€˜0) = (πΉβ€˜π‘‹))
7170adantr 481 . . . . . . . . . . . . . . . 16 ((𝑋 = 0 ∧ π‘Œ = 𝐾) β†’ (πΉβ€˜0) = (πΉβ€˜π‘‹))
7238adantl 482 . . . . . . . . . . . . . . . 16 ((𝑋 = 0 ∧ π‘Œ = 𝐾) β†’ (πΉβ€˜πΎ) = (πΉβ€˜π‘Œ))
7371, 72neeq12d 3002 . . . . . . . . . . . . . . 15 ((𝑋 = 0 ∧ π‘Œ = 𝐾) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) ↔ (πΉβ€˜π‘‹) β‰  (πΉβ€˜π‘Œ)))
74 df-ne 2941 . . . . . . . . . . . . . . . 16 ((πΉβ€˜π‘‹) β‰  (πΉβ€˜π‘Œ) ↔ Β¬ (πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ))
7574, 22sylbi 216 . . . . . . . . . . . . . . 15 ((πΉβ€˜π‘‹) β‰  (πΉβ€˜π‘Œ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ))
7673, 75syl6bi 252 . . . . . . . . . . . . . 14 ((𝑋 = 0 ∧ π‘Œ = 𝐾) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ)))
77762a1d 26 . . . . . . . . . . . . 13 ((𝑋 = 0 ∧ π‘Œ = 𝐾) β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (((𝐹 β€œ {0, 𝐾}) ∩ (𝐹 β€œ (1..^𝐾))) = βˆ… β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ)))))
78 eqtr3 2758 . . . . . . . . . . . . . 14 ((𝑋 = 𝐾 ∧ π‘Œ = 𝐾) β†’ 𝑋 = π‘Œ)
7978, 55syl 17 . . . . . . . . . . . . 13 ((𝑋 = 𝐾 ∧ π‘Œ = 𝐾) β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (((𝐹 β€œ {0, 𝐾}) ∩ (𝐹 β€œ (1..^𝐾))) = βˆ… β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ)))))
8056, 68, 77, 79ccase 1036 . . . . . . . . . . . 12 (((𝑋 = 0 ∨ 𝑋 = 𝐾) ∧ (π‘Œ = 0 ∨ π‘Œ = 𝐾)) β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (((𝐹 β€œ {0, 𝐾}) ∩ (𝐹 β€œ (1..^𝐾))) = βˆ… β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ)))))
8180ex 413 . . . . . . . . . . 11 ((𝑋 = 0 ∨ 𝑋 = 𝐾) β†’ ((π‘Œ = 0 ∨ π‘Œ = 𝐾) β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (((𝐹 β€œ {0, 𝐾}) ∩ (𝐹 β€œ (1..^𝐾))) = βˆ… β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ))))))
8252, 81syl 17 . . . . . . . . . 10 ((𝑋 ∈ (0...𝐾) ∧ Β¬ 𝑋 ∈ (1..^𝐾)) β†’ ((π‘Œ = 0 ∨ π‘Œ = 𝐾) β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (((𝐹 β€œ {0, 𝐾}) ∩ (𝐹 β€œ (1..^𝐾))) = βˆ… β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ))))))
8382expcom 414 . . . . . . . . 9 (Β¬ 𝑋 ∈ (1..^𝐾) β†’ (𝑋 ∈ (0...𝐾) β†’ ((π‘Œ = 0 ∨ π‘Œ = 𝐾) β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (((𝐹 β€œ {0, 𝐾}) ∩ (𝐹 β€œ (1..^𝐾))) = βˆ… β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ)))))))
8451, 83pm2.61i 182 . . . . . . . 8 (𝑋 ∈ (0...𝐾) β†’ ((π‘Œ = 0 ∨ π‘Œ = 𝐾) β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (((𝐹 β€œ {0, 𝐾}) ∩ (𝐹 β€œ (1..^𝐾))) = βˆ… β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ))))))
8584com12 32 . . . . . . 7 ((π‘Œ = 0 ∨ π‘Œ = 𝐾) β†’ (𝑋 ∈ (0...𝐾) β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (((𝐹 β€œ {0, 𝐾}) ∩ (𝐹 β€œ (1..^𝐾))) = βˆ… β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ))))))
861, 85syl 17 . . . . . 6 ((π‘Œ ∈ (0...𝐾) ∧ Β¬ π‘Œ ∈ (1..^𝐾)) β†’ (𝑋 ∈ (0...𝐾) β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (((𝐹 β€œ {0, 𝐾}) ∩ (𝐹 β€œ (1..^𝐾))) = βˆ… β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ))))))
8786ex 413 . . . . 5 (π‘Œ ∈ (0...𝐾) β†’ (Β¬ π‘Œ ∈ (1..^𝐾) β†’ (𝑋 ∈ (0...𝐾) β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (((𝐹 β€œ {0, 𝐾}) ∩ (𝐹 β€œ (1..^𝐾))) = βˆ… β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ)))))))
8887com23 86 . . . 4 (π‘Œ ∈ (0...𝐾) β†’ (𝑋 ∈ (0...𝐾) β†’ (Β¬ π‘Œ ∈ (1..^𝐾) β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (((𝐹 β€œ {0, 𝐾}) ∩ (𝐹 β€œ (1..^𝐾))) = βˆ… β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ)))))))
8988impcom 408 . . 3 ((𝑋 ∈ (0...𝐾) ∧ π‘Œ ∈ (0...𝐾)) β†’ (Β¬ π‘Œ ∈ (1..^𝐾) β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (((𝐹 β€œ {0, 𝐾}) ∩ (𝐹 β€œ (1..^𝐾))) = βˆ… β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ))))))
9089com12 32 . 2 (Β¬ π‘Œ ∈ (1..^𝐾) β†’ ((𝑋 ∈ (0...𝐾) ∧ π‘Œ ∈ (0...𝐾)) β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (((𝐹 β€œ {0, 𝐾}) ∩ (𝐹 β€œ (1..^𝐾))) = βˆ… β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ))))))
9190com25 99 1 (Β¬ π‘Œ ∈ (1..^𝐾) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜πΎ) β†’ ((𝐹:(0...𝐾)βŸΆπ‘‰ ∧ 𝐾 ∈ β„•0) β†’ (((𝐹 β€œ {0, 𝐾}) ∩ (𝐹 β€œ (1..^𝐾))) = βˆ… β†’ ((𝑋 ∈ (0...𝐾) ∧ π‘Œ ∈ (0...𝐾)) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑋 = π‘Œ))))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106   β‰  wne 2940   βˆ‰ wnel 3046  βˆ€wral 3061  βˆƒwrex 3070   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  {cpr 4630   β€œ cima 5679   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411  0cc0 11112  1c1 11113  β„•0cn0 12474  β„€β‰₯cuz 12824  ...cfz 13486  ..^cfzo 13629
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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-n0 12475  df-z 12561  df-uz 12825  df-fz 13487  df-fzo 13630
This theorem is referenced by:  injresinj  13755
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