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Theorem fvineqsneu 34979
 Description: A theorem about functions where the image of every point intersects the domain only at that point. (Contributed by ML, 27-Mar-2021.)
Assertion
Ref Expression
fvineqsneu ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → ∀𝑞𝐴 ∃!𝑥 ∈ ran 𝐹 𝑞𝑥)
Distinct variable groups:   𝐴,𝑞,𝑥   𝐹,𝑞,𝑥   𝐴,𝑝   𝐹,𝑝

Proof of Theorem fvineqsneu
Dummy variables 𝑜 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnfvelrn 6835 . . . . . 6 ((𝐹 Fn 𝐴𝑜𝐴) → (𝐹𝑜) ∈ ran 𝐹)
21ex 416 . . . . 5 (𝐹 Fn 𝐴 → (𝑜𝐴 → (𝐹𝑜) ∈ ran 𝐹))
32adantr 484 . . . 4 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (𝑜𝐴 → (𝐹𝑜) ∈ ran 𝐹))
4 fnrnfv 6710 . . . . . . . . . 10 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑝𝐴 𝑦 = (𝐹𝑝)})
54abeq2d 2924 . . . . . . . . 9 (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑝𝐴 𝑦 = (𝐹𝑝)))
65adantr 484 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑝𝐴 𝑦 = (𝐹𝑝)))
7 nfv 1915 . . . . . . . . . 10 𝑝 𝐹 Fn 𝐴
8 nfra1 3183 . . . . . . . . . 10 𝑝𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}
97, 8nfan 1900 . . . . . . . . 9 𝑝(𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝})
10 nfv 1915 . . . . . . . . 9 𝑝𝑜𝐴 (𝑜𝑦𝑦 = (𝐹𝑜))
11 eleq2 2878 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝐹𝑝) → (𝑜𝑦𝑜 ∈ (𝐹𝑝)))
12 elin 3899 . . . . . . . . . . . . . . . . . . . . 21 (𝑜 ∈ ((𝐹𝑝) ∩ 𝐴) ↔ (𝑜 ∈ (𝐹𝑝) ∧ 𝑜𝐴))
1312rbaib 542 . . . . . . . . . . . . . . . . . . . 20 (𝑜𝐴 → (𝑜 ∈ ((𝐹𝑝) ∩ 𝐴) ↔ 𝑜 ∈ (𝐹𝑝)))
1413ad2antll 728 . . . . . . . . . . . . . . . . . . 19 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝𝐴𝑜𝐴)) → (𝑜 ∈ ((𝐹𝑝) ∩ 𝐴) ↔ 𝑜 ∈ (𝐹𝑝)))
15 rsp 3170 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} → (𝑝𝐴 → ((𝐹𝑝) ∩ 𝐴) = {𝑝}))
16 eleq2 2878 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐹𝑝) ∩ 𝐴) = {𝑝} → (𝑜 ∈ ((𝐹𝑝) ∩ 𝐴) ↔ 𝑜 ∈ {𝑝}))
17 velsn 4544 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑜 ∈ {𝑝} ↔ 𝑜 = 𝑝)
18 equcom 2025 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑜 = 𝑝𝑝 = 𝑜)
1917, 18bitri 278 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑜 ∈ {𝑝} ↔ 𝑝 = 𝑜)
2016, 19syl6bb 290 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐹𝑝) ∩ 𝐴) = {𝑝} → (𝑜 ∈ ((𝐹𝑝) ∩ 𝐴) ↔ 𝑝 = 𝑜))
2115, 20syl6 35 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} → (𝑝𝐴 → (𝑜 ∈ ((𝐹𝑝) ∩ 𝐴) ↔ 𝑝 = 𝑜)))
2221adantl 485 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (𝑝𝐴 → (𝑜 ∈ ((𝐹𝑝) ∩ 𝐴) ↔ 𝑝 = 𝑜)))
2322adantrd 495 . . . . . . . . . . . . . . . . . . . 20 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → ((𝑝𝐴𝑜𝐴) → (𝑜 ∈ ((𝐹𝑝) ∩ 𝐴) ↔ 𝑝 = 𝑜)))
2423imp 410 . . . . . . . . . . . . . . . . . . 19 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝𝐴𝑜𝐴)) → (𝑜 ∈ ((𝐹𝑝) ∩ 𝐴) ↔ 𝑝 = 𝑜))
2514, 24bitr3d 284 . . . . . . . . . . . . . . . . . 18 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝𝐴𝑜𝐴)) → (𝑜 ∈ (𝐹𝑝) ↔ 𝑝 = 𝑜))
2611, 25sylan9bbr 514 . . . . . . . . . . . . . . . . 17 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝𝐴𝑜𝐴)) ∧ 𝑦 = (𝐹𝑝)) → (𝑜𝑦𝑝 = 𝑜))
2726ex 416 . . . . . . . . . . . . . . . 16 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝𝐴𝑜𝐴)) → (𝑦 = (𝐹𝑝) → (𝑜𝑦𝑝 = 𝑜)))
2827anass1rs 654 . . . . . . . . . . . . . . 15 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑜𝐴) ∧ 𝑝𝐴) → (𝑦 = (𝐹𝑝) → (𝑜𝑦𝑝 = 𝑜)))
2928impr 458 . . . . . . . . . . . . . 14 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑜𝐴) ∧ (𝑝𝐴𝑦 = (𝐹𝑝))) → (𝑜𝑦𝑝 = 𝑜))
3029an32s 651 . . . . . . . . . . . . 13 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝𝐴𝑦 = (𝐹𝑝))) ∧ 𝑜𝐴) → (𝑜𝑦𝑝 = 𝑜))
31 eqeq1 2802 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝐹𝑝) → (𝑦 = (𝐹𝑜) ↔ (𝐹𝑝) = (𝐹𝑜)))
32 dffn3 6507 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
33 fvineqsnf1 34978 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐹:𝐴⟶ran 𝐹 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → 𝐹:𝐴1-1→ran 𝐹)
3432, 33sylanb 584 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → 𝐹:𝐴1-1→ran 𝐹)
35 dff13 7001 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐹:𝐴1-1→ran 𝐹 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ∀𝑝𝐴𝑜𝐴 ((𝐹𝑝) = (𝐹𝑜) → 𝑝 = 𝑜)))
3634, 35sylib 221 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (𝐹:𝐴⟶ran 𝐹 ∧ ∀𝑝𝐴𝑜𝐴 ((𝐹𝑝) = (𝐹𝑜) → 𝑝 = 𝑜)))
3736simprd 499 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → ∀𝑝𝐴𝑜𝐴 ((𝐹𝑝) = (𝐹𝑜) → 𝑝 = 𝑜))
38 rsp 3170 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑝𝐴𝑜𝐴 ((𝐹𝑝) = (𝐹𝑜) → 𝑝 = 𝑜) → (𝑝𝐴 → ∀𝑜𝐴 ((𝐹𝑝) = (𝐹𝑜) → 𝑝 = 𝑜)))
3937, 38syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (𝑝𝐴 → ∀𝑜𝐴 ((𝐹𝑝) = (𝐹𝑜) → 𝑝 = 𝑜)))
40 rsp 3170 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑜𝐴 ((𝐹𝑝) = (𝐹𝑜) → 𝑝 = 𝑜) → (𝑜𝐴 → ((𝐹𝑝) = (𝐹𝑜) → 𝑝 = 𝑜)))
4139, 40syl6 35 . . . . . . . . . . . . . . . . . . . 20 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (𝑝𝐴 → (𝑜𝐴 → ((𝐹𝑝) = (𝐹𝑜) → 𝑝 = 𝑜))))
4241imp32 422 . . . . . . . . . . . . . . . . . . 19 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝𝐴𝑜𝐴)) → ((𝐹𝑝) = (𝐹𝑜) → 𝑝 = 𝑜))
43 fveq2 6655 . . . . . . . . . . . . . . . . . . 19 (𝑝 = 𝑜 → (𝐹𝑝) = (𝐹𝑜))
4442, 43impbid1 228 . . . . . . . . . . . . . . . . . 18 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝𝐴𝑜𝐴)) → ((𝐹𝑝) = (𝐹𝑜) ↔ 𝑝 = 𝑜))
4531, 44sylan9bbr 514 . . . . . . . . . . . . . . . . 17 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝𝐴𝑜𝐴)) ∧ 𝑦 = (𝐹𝑝)) → (𝑦 = (𝐹𝑜) ↔ 𝑝 = 𝑜))
4645ex 416 . . . . . . . . . . . . . . . 16 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝𝐴𝑜𝐴)) → (𝑦 = (𝐹𝑝) → (𝑦 = (𝐹𝑜) ↔ 𝑝 = 𝑜)))
4746anass1rs 654 . . . . . . . . . . . . . . 15 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑜𝐴) ∧ 𝑝𝐴) → (𝑦 = (𝐹𝑝) → (𝑦 = (𝐹𝑜) ↔ 𝑝 = 𝑜)))
4847impr 458 . . . . . . . . . . . . . 14 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑜𝐴) ∧ (𝑝𝐴𝑦 = (𝐹𝑝))) → (𝑦 = (𝐹𝑜) ↔ 𝑝 = 𝑜))
4948an32s 651 . . . . . . . . . . . . 13 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝𝐴𝑦 = (𝐹𝑝))) ∧ 𝑜𝐴) → (𝑦 = (𝐹𝑜) ↔ 𝑝 = 𝑜))
5030, 49bitr4d 285 . . . . . . . . . . . 12 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝𝐴𝑦 = (𝐹𝑝))) ∧ 𝑜𝐴) → (𝑜𝑦𝑦 = (𝐹𝑜)))
5150ex 416 . . . . . . . . . . 11 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝𝐴𝑦 = (𝐹𝑝))) → (𝑜𝐴 → (𝑜𝑦𝑦 = (𝐹𝑜))))
5251ralrimiv 3148 . . . . . . . . . 10 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝𝐴𝑦 = (𝐹𝑝))) → ∀𝑜𝐴 (𝑜𝑦𝑦 = (𝐹𝑜)))
5352exp32 424 . . . . . . . . 9 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (𝑝𝐴 → (𝑦 = (𝐹𝑝) → ∀𝑜𝐴 (𝑜𝑦𝑦 = (𝐹𝑜)))))
549, 10, 53rexlimd 3277 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (∃𝑝𝐴 𝑦 = (𝐹𝑝) → ∀𝑜𝐴 (𝑜𝑦𝑦 = (𝐹𝑜))))
556, 54sylbid 243 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (𝑦 ∈ ran 𝐹 → ∀𝑜𝐴 (𝑜𝑦𝑦 = (𝐹𝑜))))
56 rsp 3170 . . . . . . 7 (∀𝑜𝐴 (𝑜𝑦𝑦 = (𝐹𝑜)) → (𝑜𝐴 → (𝑜𝑦𝑦 = (𝐹𝑜))))
5755, 56syl6 35 . . . . . 6 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (𝑦 ∈ ran 𝐹 → (𝑜𝐴 → (𝑜𝑦𝑦 = (𝐹𝑜)))))
5857com23 86 . . . . 5 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (𝑜𝐴 → (𝑦 ∈ ran 𝐹 → (𝑜𝑦𝑦 = (𝐹𝑜)))))
5958ralrimdv 3153 . . . 4 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (𝑜𝐴 → ∀𝑦 ∈ ran 𝐹(𝑜𝑦𝑦 = (𝐹𝑜))))
60 reu6i 3669 . . . . 5 (((𝐹𝑜) ∈ ran 𝐹 ∧ ∀𝑦 ∈ ran 𝐹(𝑜𝑦𝑦 = (𝐹𝑜))) → ∃!𝑦 ∈ ran 𝐹 𝑜𝑦)
6160ex 416 . . . 4 ((𝐹𝑜) ∈ ran 𝐹 → (∀𝑦 ∈ ran 𝐹(𝑜𝑦𝑦 = (𝐹𝑜)) → ∃!𝑦 ∈ ran 𝐹 𝑜𝑦))
623, 59, 61syl6c 70 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (𝑜𝐴 → ∃!𝑦 ∈ ran 𝐹 𝑜𝑦))
6362ralrimiv 3148 . 2 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → ∀𝑜𝐴 ∃!𝑦 ∈ ran 𝐹 𝑜𝑦)
64 nfv 1915 . . . 4 𝑥 𝑞 = 𝑜
65 nfv 1915 . . . 4 𝑦 𝑞 = 𝑜
66 nfvd 1916 . . . 4 (𝑞 = 𝑜 → Ⅎ𝑦 𝑞𝑥)
67 nfvd 1916 . . . 4 (𝑞 = 𝑜 → Ⅎ𝑥 𝑜𝑦)
68 eleq12 2879 . . . . 5 ((𝑞 = 𝑜𝑥 = 𝑦) → (𝑞𝑥𝑜𝑦))
6968ex 416 . . . 4 (𝑞 = 𝑜 → (𝑥 = 𝑦 → (𝑞𝑥𝑜𝑦)))
7064, 65, 66, 67, 69cbvreud 34941 . . 3 (𝑞 = 𝑜 → (∃!𝑥 ∈ ran 𝐹 𝑞𝑥 ↔ ∃!𝑦 ∈ ran 𝐹 𝑜𝑦))
7170cbvralvw 3397 . 2 (∀𝑞𝐴 ∃!𝑥 ∈ ran 𝐹 𝑞𝑥 ↔ ∀𝑜𝐴 ∃!𝑦 ∈ ran 𝐹 𝑜𝑦)
7263, 71sylibr 237 1 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → ∀𝑞𝐴 ∃!𝑥 ∈ ran 𝐹 𝑞𝑥)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  ∃!wreu 3108   ∩ cin 3882  {csn 4528  ran crn 5524   Fn wfn 6327  ⟶wf 6328  –1-1→wf1 6329  ‘cfv 6332 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fv 6340 This theorem is referenced by:  fvineqsneq  34980
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