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Theorem fvineqsneu 37021
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 7089 . . . . . 6 ((𝐹 Fn 𝐴𝑜𝐴) → (𝐹𝑜) ∈ ran 𝐹)
21ex 411 . . . . 5 (𝐹 Fn 𝐴 → (𝑜𝐴 → (𝐹𝑜) ∈ ran 𝐹))
32adantr 479 . . . 4 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (𝑜𝐴 → (𝐹𝑜) ∈ ran 𝐹))
4 fnrnfv 6957 . . . . . . . . . 10 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑝𝐴 𝑦 = (𝐹𝑝)})
54eqabrd 2868 . . . . . . . . 9 (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑝𝐴 𝑦 = (𝐹𝑝)))
65adantr 479 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑝𝐴 𝑦 = (𝐹𝑝)))
7 nfv 1909 . . . . . . . . . 10 𝑝 𝐹 Fn 𝐴
8 nfra1 3271 . . . . . . . . . 10 𝑝𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}
97, 8nfan 1894 . . . . . . . . 9 𝑝(𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝})
10 nfv 1909 . . . . . . . . 9 𝑝𝑜𝐴 (𝑜𝑦𝑦 = (𝐹𝑜))
11 eleq2 2814 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝐹𝑝) → (𝑜𝑦𝑜 ∈ (𝐹𝑝)))
12 elin 3960 . . . . . . . . . . . . . . . . . . . . 21 (𝑜 ∈ ((𝐹𝑝) ∩ 𝐴) ↔ (𝑜 ∈ (𝐹𝑝) ∧ 𝑜𝐴))
1312rbaib 537 . . . . . . . . . . . . . . . . . . . 20 (𝑜𝐴 → (𝑜 ∈ ((𝐹𝑝) ∩ 𝐴) ↔ 𝑜 ∈ (𝐹𝑝)))
1413ad2antll 727 . . . . . . . . . . . . . . . . . . 19 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝𝐴𝑜𝐴)) → (𝑜 ∈ ((𝐹𝑝) ∩ 𝐴) ↔ 𝑜 ∈ (𝐹𝑝)))
15 rsp 3234 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} → (𝑝𝐴 → ((𝐹𝑝) ∩ 𝐴) = {𝑝}))
16 eleq2 2814 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐹𝑝) ∩ 𝐴) = {𝑝} → (𝑜 ∈ ((𝐹𝑝) ∩ 𝐴) ↔ 𝑜 ∈ {𝑝}))
17 velsn 4646 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑜 ∈ {𝑝} ↔ 𝑜 = 𝑝)
18 equcom 2013 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑜 = 𝑝𝑝 = 𝑜)
1917, 18bitri 274 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑜 ∈ {𝑝} ↔ 𝑝 = 𝑜)
2016, 19bitrdi 286 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐹𝑝) ∩ 𝐴) = {𝑝} → (𝑜 ∈ ((𝐹𝑝) ∩ 𝐴) ↔ 𝑝 = 𝑜))
2115, 20syl6 35 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} → (𝑝𝐴 → (𝑜 ∈ ((𝐹𝑝) ∩ 𝐴) ↔ 𝑝 = 𝑜)))
2221adantl 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (𝑝𝐴 → (𝑜 ∈ ((𝐹𝑝) ∩ 𝐴) ↔ 𝑝 = 𝑜)))
2322adantrd 490 . . . . . . . . . . . . . . . . . . . 20 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → ((𝑝𝐴𝑜𝐴) → (𝑜 ∈ ((𝐹𝑝) ∩ 𝐴) ↔ 𝑝 = 𝑜)))
2423imp 405 . . . . . . . . . . . . . . . . . . 19 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝𝐴𝑜𝐴)) → (𝑜 ∈ ((𝐹𝑝) ∩ 𝐴) ↔ 𝑝 = 𝑜))
2514, 24bitr3d 280 . . . . . . . . . . . . . . . . . 18 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝𝐴𝑜𝐴)) → (𝑜 ∈ (𝐹𝑝) ↔ 𝑝 = 𝑜))
2611, 25sylan9bbr 509 . . . . . . . . . . . . . . . . 17 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝𝐴𝑜𝐴)) ∧ 𝑦 = (𝐹𝑝)) → (𝑜𝑦𝑝 = 𝑜))
2726ex 411 . . . . . . . . . . . . . . . 16 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝𝐴𝑜𝐴)) → (𝑦 = (𝐹𝑝) → (𝑜𝑦𝑝 = 𝑜)))
2827anass1rs 653 . . . . . . . . . . . . . . 15 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑜𝐴) ∧ 𝑝𝐴) → (𝑦 = (𝐹𝑝) → (𝑜𝑦𝑝 = 𝑜)))
2928impr 453 . . . . . . . . . . . . . 14 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑜𝐴) ∧ (𝑝𝐴𝑦 = (𝐹𝑝))) → (𝑜𝑦𝑝 = 𝑜))
3029an32s 650 . . . . . . . . . . . . 13 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝𝐴𝑦 = (𝐹𝑝))) ∧ 𝑜𝐴) → (𝑜𝑦𝑝 = 𝑜))
31 eqeq1 2729 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝐹𝑝) → (𝑦 = (𝐹𝑜) ↔ (𝐹𝑝) = (𝐹𝑜)))
32 dffn3 6735 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
33 fvineqsnf1 37020 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐹:𝐴⟶ran 𝐹 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → 𝐹:𝐴1-1→ran 𝐹)
3432, 33sylanb 579 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → 𝐹:𝐴1-1→ran 𝐹)
35 dff13 7265 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐹:𝐴1-1→ran 𝐹 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ∀𝑝𝐴𝑜𝐴 ((𝐹𝑝) = (𝐹𝑜) → 𝑝 = 𝑜)))
3634, 35sylib 217 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (𝐹:𝐴⟶ran 𝐹 ∧ ∀𝑝𝐴𝑜𝐴 ((𝐹𝑝) = (𝐹𝑜) → 𝑝 = 𝑜)))
3736simprd 494 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → ∀𝑝𝐴𝑜𝐴 ((𝐹𝑝) = (𝐹𝑜) → 𝑝 = 𝑜))
38 rsp 3234 . . . . . . . . . . . . . . . . . . . . . 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 3234 . . . . . . . . . . . . . . . . . . . . 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 417 . . . . . . . . . . . . . . . . . . 19 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝𝐴𝑜𝐴)) → ((𝐹𝑝) = (𝐹𝑜) → 𝑝 = 𝑜))
43 fveq2 6896 . . . . . . . . . . . . . . . . . . 19 (𝑝 = 𝑜 → (𝐹𝑝) = (𝐹𝑜))
4442, 43impbid1 224 . . . . . . . . . . . . . . . . . 18 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝𝐴𝑜𝐴)) → ((𝐹𝑝) = (𝐹𝑜) ↔ 𝑝 = 𝑜))
4531, 44sylan9bbr 509 . . . . . . . . . . . . . . . . 17 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝𝐴𝑜𝐴)) ∧ 𝑦 = (𝐹𝑝)) → (𝑦 = (𝐹𝑜) ↔ 𝑝 = 𝑜))
4645ex 411 . . . . . . . . . . . . . . . 16 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝𝐴𝑜𝐴)) → (𝑦 = (𝐹𝑝) → (𝑦 = (𝐹𝑜) ↔ 𝑝 = 𝑜)))
4746anass1rs 653 . . . . . . . . . . . . . . 15 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑜𝐴) ∧ 𝑝𝐴) → (𝑦 = (𝐹𝑝) → (𝑦 = (𝐹𝑜) ↔ 𝑝 = 𝑜)))
4847impr 453 . . . . . . . . . . . . . 14 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑜𝐴) ∧ (𝑝𝐴𝑦 = (𝐹𝑝))) → (𝑦 = (𝐹𝑜) ↔ 𝑝 = 𝑜))
4948an32s 650 . . . . . . . . . . . . 13 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝𝐴𝑦 = (𝐹𝑝))) ∧ 𝑜𝐴) → (𝑦 = (𝐹𝑜) ↔ 𝑝 = 𝑜))
5030, 49bitr4d 281 . . . . . . . . . . . 12 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝𝐴𝑦 = (𝐹𝑝))) ∧ 𝑜𝐴) → (𝑜𝑦𝑦 = (𝐹𝑜)))
5150ex 411 . . . . . . . . . . 11 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝𝐴𝑦 = (𝐹𝑝))) → (𝑜𝐴 → (𝑜𝑦𝑦 = (𝐹𝑜))))
5251ralrimiv 3134 . . . . . . . . . 10 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝𝐴𝑦 = (𝐹𝑝))) → ∀𝑜𝐴 (𝑜𝑦𝑦 = (𝐹𝑜)))
5352exp32 419 . . . . . . . . 9 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (𝑝𝐴 → (𝑦 = (𝐹𝑝) → ∀𝑜𝐴 (𝑜𝑦𝑦 = (𝐹𝑜)))))
549, 10, 53rexlimd 3253 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (∃𝑝𝐴 𝑦 = (𝐹𝑝) → ∀𝑜𝐴 (𝑜𝑦𝑦 = (𝐹𝑜))))
556, 54sylbid 239 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (𝑦 ∈ ran 𝐹 → ∀𝑜𝐴 (𝑜𝑦𝑦 = (𝐹𝑜))))
56 rsp 3234 . . . . . . 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 3141 . . . 4 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (𝑜𝐴 → ∀𝑦 ∈ ran 𝐹(𝑜𝑦𝑦 = (𝐹𝑜))))
60 reu6i 3720 . . . . 5 (((𝐹𝑜) ∈ ran 𝐹 ∧ ∀𝑦 ∈ ran 𝐹(𝑜𝑦𝑦 = (𝐹𝑜))) → ∃!𝑦 ∈ ran 𝐹 𝑜𝑦)
6160ex 411 . . . 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 3134 . 2 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → ∀𝑜𝐴 ∃!𝑦 ∈ ran 𝐹 𝑜𝑦)
64 nfv 1909 . . . 4 𝑥 𝑞 = 𝑜
65 nfv 1909 . . . 4 𝑦 𝑞 = 𝑜
66 nfvd 1910 . . . 4 (𝑞 = 𝑜 → Ⅎ𝑦 𝑞𝑥)
67 nfvd 1910 . . . 4 (𝑞 = 𝑜 → Ⅎ𝑥 𝑜𝑦)
68 eleq12 2815 . . . . 5 ((𝑞 = 𝑜𝑥 = 𝑦) → (𝑞𝑥𝑜𝑦))
6968ex 411 . . . 4 (𝑞 = 𝑜 → (𝑥 = 𝑦 → (𝑞𝑥𝑜𝑦)))
7064, 65, 66, 67, 69cbvreud 36983 . . 3 (𝑞 = 𝑜 → (∃!𝑥 ∈ ran 𝐹 𝑞𝑥 ↔ ∃!𝑦 ∈ ran 𝐹 𝑜𝑦))
7170cbvralvw 3224 . 2 (∀𝑞𝐴 ∃!𝑥 ∈ ran 𝐹 𝑞𝑥 ↔ ∀𝑜𝐴 ∃!𝑦 ∈ ran 𝐹 𝑜𝑦)
7263, 71sylibr 233 1 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → ∀𝑞𝐴 ∃!𝑥 ∈ ran 𝐹 𝑞𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3050  wrex 3059  ∃!wreu 3361  cin 3943  {csn 4630  ran crn 5679   Fn wfn 6544  wf 6545  1-1wf1 6546  cfv 6549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fv 6557
This theorem is referenced by:  fvineqsneq  37022
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