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Theorem fvineqsneu 35911
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 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) β†’ βˆ€π‘ž ∈ 𝐴 βˆƒ!π‘₯ ∈ ran 𝐹 π‘ž ∈ π‘₯)
Distinct variable groups:   𝐴,π‘ž,π‘₯   𝐹,π‘ž,π‘₯   𝐴,𝑝   𝐹,𝑝

Proof of Theorem fvineqsneu
Dummy variables π‘œ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnfvelrn 7036 . . . . . 6 ((𝐹 Fn 𝐴 ∧ π‘œ ∈ 𝐴) β†’ (πΉβ€˜π‘œ) ∈ ran 𝐹)
21ex 414 . . . . 5 (𝐹 Fn 𝐴 β†’ (π‘œ ∈ 𝐴 β†’ (πΉβ€˜π‘œ) ∈ ran 𝐹))
32adantr 482 . . . 4 ((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) β†’ (π‘œ ∈ 𝐴 β†’ (πΉβ€˜π‘œ) ∈ ran 𝐹))
4 fnrnfv 6907 . . . . . . . . . 10 (𝐹 Fn 𝐴 β†’ ran 𝐹 = {𝑦 ∣ βˆƒπ‘ ∈ 𝐴 𝑦 = (πΉβ€˜π‘)})
54eqabd 2881 . . . . . . . . 9 (𝐹 Fn 𝐴 β†’ (𝑦 ∈ ran 𝐹 ↔ βˆƒπ‘ ∈ 𝐴 𝑦 = (πΉβ€˜π‘)))
65adantr 482 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) β†’ (𝑦 ∈ ran 𝐹 ↔ βˆƒπ‘ ∈ 𝐴 𝑦 = (πΉβ€˜π‘)))
7 nfv 1918 . . . . . . . . . 10 Ⅎ𝑝 𝐹 Fn 𝐴
8 nfra1 3270 . . . . . . . . . 10 β„²π‘βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}
97, 8nfan 1903 . . . . . . . . 9 Ⅎ𝑝(𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝})
10 nfv 1918 . . . . . . . . 9 β„²π‘βˆ€π‘œ ∈ 𝐴 (π‘œ ∈ 𝑦 ↔ 𝑦 = (πΉβ€˜π‘œ))
11 eleq2 2827 . . . . . . . . . . . . . . . . . 18 (𝑦 = (πΉβ€˜π‘) β†’ (π‘œ ∈ 𝑦 ↔ π‘œ ∈ (πΉβ€˜π‘)))
12 elin 3931 . . . . . . . . . . . . . . . . . . . . 21 (π‘œ ∈ ((πΉβ€˜π‘) ∩ 𝐴) ↔ (π‘œ ∈ (πΉβ€˜π‘) ∧ π‘œ ∈ 𝐴))
1312rbaib 540 . . . . . . . . . . . . . . . . . . . 20 (π‘œ ∈ 𝐴 β†’ (π‘œ ∈ ((πΉβ€˜π‘) ∩ 𝐴) ↔ π‘œ ∈ (πΉβ€˜π‘)))
1413ad2antll 728 . . . . . . . . . . . . . . . . . . 19 (((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ π‘œ ∈ 𝐴)) β†’ (π‘œ ∈ ((πΉβ€˜π‘) ∩ 𝐴) ↔ π‘œ ∈ (πΉβ€˜π‘)))
15 rsp 3233 . . . . . . . . . . . . . . . . . . . . . . 23 (βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝} β†’ (𝑝 ∈ 𝐴 β†’ ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}))
16 eleq2 2827 . . . . . . . . . . . . . . . . . . . . . . . 24 (((πΉβ€˜π‘) ∩ 𝐴) = {𝑝} β†’ (π‘œ ∈ ((πΉβ€˜π‘) ∩ 𝐴) ↔ π‘œ ∈ {𝑝}))
17 velsn 4607 . . . . . . . . . . . . . . . . . . . . . . . . 25 (π‘œ ∈ {𝑝} ↔ π‘œ = 𝑝)
18 equcom 2022 . . . . . . . . . . . . . . . . . . . . . . . . 25 (π‘œ = 𝑝 ↔ 𝑝 = π‘œ)
1917, 18bitri 275 . . . . . . . . . . . . . . . . . . . . . . . 24 (π‘œ ∈ {𝑝} ↔ 𝑝 = π‘œ)
2016, 19bitrdi 287 . . . . . . . . . . . . . . . . . . . . . . 23 (((πΉβ€˜π‘) ∩ 𝐴) = {𝑝} β†’ (π‘œ ∈ ((πΉβ€˜π‘) ∩ 𝐴) ↔ 𝑝 = π‘œ))
2115, 20syl6 35 . . . . . . . . . . . . . . . . . . . . . 22 (βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝} β†’ (𝑝 ∈ 𝐴 β†’ (π‘œ ∈ ((πΉβ€˜π‘) ∩ 𝐴) ↔ 𝑝 = π‘œ)))
2221adantl 483 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) β†’ (𝑝 ∈ 𝐴 β†’ (π‘œ ∈ ((πΉβ€˜π‘) ∩ 𝐴) ↔ 𝑝 = π‘œ)))
2322adantrd 493 . . . . . . . . . . . . . . . . . . . 20 ((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) β†’ ((𝑝 ∈ 𝐴 ∧ π‘œ ∈ 𝐴) β†’ (π‘œ ∈ ((πΉβ€˜π‘) ∩ 𝐴) ↔ 𝑝 = π‘œ)))
2423imp 408 . . . . . . . . . . . . . . . . . . 19 (((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ π‘œ ∈ 𝐴)) β†’ (π‘œ ∈ ((πΉβ€˜π‘) ∩ 𝐴) ↔ 𝑝 = π‘œ))
2514, 24bitr3d 281 . . . . . . . . . . . . . . . . . 18 (((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ π‘œ ∈ 𝐴)) β†’ (π‘œ ∈ (πΉβ€˜π‘) ↔ 𝑝 = π‘œ))
2611, 25sylan9bbr 512 . . . . . . . . . . . . . . . . 17 ((((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ π‘œ ∈ 𝐴)) ∧ 𝑦 = (πΉβ€˜π‘)) β†’ (π‘œ ∈ 𝑦 ↔ 𝑝 = π‘œ))
2726ex 414 . . . . . . . . . . . . . . . 16 (((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ π‘œ ∈ 𝐴)) β†’ (𝑦 = (πΉβ€˜π‘) β†’ (π‘œ ∈ 𝑦 ↔ 𝑝 = π‘œ)))
2827anass1rs 654 . . . . . . . . . . . . . . 15 ((((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) ∧ π‘œ ∈ 𝐴) ∧ 𝑝 ∈ 𝐴) β†’ (𝑦 = (πΉβ€˜π‘) β†’ (π‘œ ∈ 𝑦 ↔ 𝑝 = π‘œ)))
2928impr 456 . . . . . . . . . . . . . 14 ((((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) ∧ π‘œ ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑦 = (πΉβ€˜π‘))) β†’ (π‘œ ∈ 𝑦 ↔ 𝑝 = π‘œ))
3029an32s 651 . . . . . . . . . . . . 13 ((((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑦 = (πΉβ€˜π‘))) ∧ π‘œ ∈ 𝐴) β†’ (π‘œ ∈ 𝑦 ↔ 𝑝 = π‘œ))
31 eqeq1 2741 . . . . . . . . . . . . . . . . . 18 (𝑦 = (πΉβ€˜π‘) β†’ (𝑦 = (πΉβ€˜π‘œ) ↔ (πΉβ€˜π‘) = (πΉβ€˜π‘œ)))
32 dffn3 6686 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟢ran 𝐹)
33 fvineqsnf1 35910 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐹:𝐴⟢ran 𝐹 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) β†’ 𝐹:𝐴–1-1β†’ran 𝐹)
3432, 33sylanb 582 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) β†’ 𝐹:𝐴–1-1β†’ran 𝐹)
35 dff13 7207 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐹:𝐴–1-1β†’ran 𝐹 ↔ (𝐹:𝐴⟢ran 𝐹 ∧ βˆ€π‘ ∈ 𝐴 βˆ€π‘œ ∈ 𝐴 ((πΉβ€˜π‘) = (πΉβ€˜π‘œ) β†’ 𝑝 = π‘œ)))
3634, 35sylib 217 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) β†’ (𝐹:𝐴⟢ran 𝐹 ∧ βˆ€π‘ ∈ 𝐴 βˆ€π‘œ ∈ 𝐴 ((πΉβ€˜π‘) = (πΉβ€˜π‘œ) β†’ 𝑝 = π‘œ)))
3736simprd 497 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) β†’ βˆ€π‘ ∈ 𝐴 βˆ€π‘œ ∈ 𝐴 ((πΉβ€˜π‘) = (πΉβ€˜π‘œ) β†’ 𝑝 = π‘œ))
38 rsp 3233 . . . . . . . . . . . . . . . . . . . . . 22 (βˆ€π‘ ∈ 𝐴 βˆ€π‘œ ∈ 𝐴 ((πΉβ€˜π‘) = (πΉβ€˜π‘œ) β†’ 𝑝 = π‘œ) β†’ (𝑝 ∈ 𝐴 β†’ βˆ€π‘œ ∈ 𝐴 ((πΉβ€˜π‘) = (πΉβ€˜π‘œ) β†’ 𝑝 = π‘œ)))
3937, 38syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) β†’ (𝑝 ∈ 𝐴 β†’ βˆ€π‘œ ∈ 𝐴 ((πΉβ€˜π‘) = (πΉβ€˜π‘œ) β†’ 𝑝 = π‘œ)))
40 rsp 3233 . . . . . . . . . . . . . . . . . . . . 21 (βˆ€π‘œ ∈ 𝐴 ((πΉβ€˜π‘) = (πΉβ€˜π‘œ) β†’ 𝑝 = π‘œ) β†’ (π‘œ ∈ 𝐴 β†’ ((πΉβ€˜π‘) = (πΉβ€˜π‘œ) β†’ 𝑝 = π‘œ)))
4139, 40syl6 35 . . . . . . . . . . . . . . . . . . . 20 ((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) β†’ (𝑝 ∈ 𝐴 β†’ (π‘œ ∈ 𝐴 β†’ ((πΉβ€˜π‘) = (πΉβ€˜π‘œ) β†’ 𝑝 = π‘œ))))
4241imp32 420 . . . . . . . . . . . . . . . . . . 19 (((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ π‘œ ∈ 𝐴)) β†’ ((πΉβ€˜π‘) = (πΉβ€˜π‘œ) β†’ 𝑝 = π‘œ))
43 fveq2 6847 . . . . . . . . . . . . . . . . . . 19 (𝑝 = π‘œ β†’ (πΉβ€˜π‘) = (πΉβ€˜π‘œ))
4442, 43impbid1 224 . . . . . . . . . . . . . . . . . 18 (((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ π‘œ ∈ 𝐴)) β†’ ((πΉβ€˜π‘) = (πΉβ€˜π‘œ) ↔ 𝑝 = π‘œ))
4531, 44sylan9bbr 512 . . . . . . . . . . . . . . . . 17 ((((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ π‘œ ∈ 𝐴)) ∧ 𝑦 = (πΉβ€˜π‘)) β†’ (𝑦 = (πΉβ€˜π‘œ) ↔ 𝑝 = π‘œ))
4645ex 414 . . . . . . . . . . . . . . . 16 (((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ π‘œ ∈ 𝐴)) β†’ (𝑦 = (πΉβ€˜π‘) β†’ (𝑦 = (πΉβ€˜π‘œ) ↔ 𝑝 = π‘œ)))
4746anass1rs 654 . . . . . . . . . . . . . . 15 ((((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) ∧ π‘œ ∈ 𝐴) ∧ 𝑝 ∈ 𝐴) β†’ (𝑦 = (πΉβ€˜π‘) β†’ (𝑦 = (πΉβ€˜π‘œ) ↔ 𝑝 = π‘œ)))
4847impr 456 . . . . . . . . . . . . . 14 ((((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) ∧ π‘œ ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑦 = (πΉβ€˜π‘))) β†’ (𝑦 = (πΉβ€˜π‘œ) ↔ 𝑝 = π‘œ))
4948an32s 651 . . . . . . . . . . . . 13 ((((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑦 = (πΉβ€˜π‘))) ∧ π‘œ ∈ 𝐴) β†’ (𝑦 = (πΉβ€˜π‘œ) ↔ 𝑝 = π‘œ))
5030, 49bitr4d 282 . . . . . . . . . . . 12 ((((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑦 = (πΉβ€˜π‘))) ∧ π‘œ ∈ 𝐴) β†’ (π‘œ ∈ 𝑦 ↔ 𝑦 = (πΉβ€˜π‘œ)))
5150ex 414 . . . . . . . . . . 11 (((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑦 = (πΉβ€˜π‘))) β†’ (π‘œ ∈ 𝐴 β†’ (π‘œ ∈ 𝑦 ↔ 𝑦 = (πΉβ€˜π‘œ))))
5251ralrimiv 3143 . . . . . . . . . 10 (((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑦 = (πΉβ€˜π‘))) β†’ βˆ€π‘œ ∈ 𝐴 (π‘œ ∈ 𝑦 ↔ 𝑦 = (πΉβ€˜π‘œ)))
5352exp32 422 . . . . . . . . 9 ((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) β†’ (𝑝 ∈ 𝐴 β†’ (𝑦 = (πΉβ€˜π‘) β†’ βˆ€π‘œ ∈ 𝐴 (π‘œ ∈ 𝑦 ↔ 𝑦 = (πΉβ€˜π‘œ)))))
549, 10, 53rexlimd 3252 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) β†’ (βˆƒπ‘ ∈ 𝐴 𝑦 = (πΉβ€˜π‘) β†’ βˆ€π‘œ ∈ 𝐴 (π‘œ ∈ 𝑦 ↔ 𝑦 = (πΉβ€˜π‘œ))))
556, 54sylbid 239 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) β†’ (𝑦 ∈ ran 𝐹 β†’ βˆ€π‘œ ∈ 𝐴 (π‘œ ∈ 𝑦 ↔ 𝑦 = (πΉβ€˜π‘œ))))
56 rsp 3233 . . . . . . 7 (βˆ€π‘œ ∈ 𝐴 (π‘œ ∈ 𝑦 ↔ 𝑦 = (πΉβ€˜π‘œ)) β†’ (π‘œ ∈ 𝐴 β†’ (π‘œ ∈ 𝑦 ↔ 𝑦 = (πΉβ€˜π‘œ))))
5755, 56syl6 35 . . . . . 6 ((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) β†’ (𝑦 ∈ ran 𝐹 β†’ (π‘œ ∈ 𝐴 β†’ (π‘œ ∈ 𝑦 ↔ 𝑦 = (πΉβ€˜π‘œ)))))
5857com23 86 . . . . 5 ((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) β†’ (π‘œ ∈ 𝐴 β†’ (𝑦 ∈ ran 𝐹 β†’ (π‘œ ∈ 𝑦 ↔ 𝑦 = (πΉβ€˜π‘œ)))))
5958ralrimdv 3150 . . . 4 ((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) β†’ (π‘œ ∈ 𝐴 β†’ βˆ€π‘¦ ∈ ran 𝐹(π‘œ ∈ 𝑦 ↔ 𝑦 = (πΉβ€˜π‘œ))))
60 reu6i 3691 . . . . 5 (((πΉβ€˜π‘œ) ∈ ran 𝐹 ∧ βˆ€π‘¦ ∈ ran 𝐹(π‘œ ∈ 𝑦 ↔ 𝑦 = (πΉβ€˜π‘œ))) β†’ βˆƒ!𝑦 ∈ ran 𝐹 π‘œ ∈ 𝑦)
6160ex 414 . . . 4 ((πΉβ€˜π‘œ) ∈ ran 𝐹 β†’ (βˆ€π‘¦ ∈ ran 𝐹(π‘œ ∈ 𝑦 ↔ 𝑦 = (πΉβ€˜π‘œ)) β†’ βˆƒ!𝑦 ∈ ran 𝐹 π‘œ ∈ 𝑦))
623, 59, 61syl6c 70 . . 3 ((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) β†’ (π‘œ ∈ 𝐴 β†’ βˆƒ!𝑦 ∈ ran 𝐹 π‘œ ∈ 𝑦))
6362ralrimiv 3143 . 2 ((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) β†’ βˆ€π‘œ ∈ 𝐴 βˆƒ!𝑦 ∈ ran 𝐹 π‘œ ∈ 𝑦)
64 nfv 1918 . . . 4 β„²π‘₯ π‘ž = π‘œ
65 nfv 1918 . . . 4 Ⅎ𝑦 π‘ž = π‘œ
66 nfvd 1919 . . . 4 (π‘ž = π‘œ β†’ Ⅎ𝑦 π‘ž ∈ π‘₯)
67 nfvd 1919 . . . 4 (π‘ž = π‘œ β†’ β„²π‘₯ π‘œ ∈ 𝑦)
68 eleq12 2828 . . . . 5 ((π‘ž = π‘œ ∧ π‘₯ = 𝑦) β†’ (π‘ž ∈ π‘₯ ↔ π‘œ ∈ 𝑦))
6968ex 414 . . . 4 (π‘ž = π‘œ β†’ (π‘₯ = 𝑦 β†’ (π‘ž ∈ π‘₯ ↔ π‘œ ∈ 𝑦)))
7064, 65, 66, 67, 69cbvreud 35873 . . 3 (π‘ž = π‘œ β†’ (βˆƒ!π‘₯ ∈ ran 𝐹 π‘ž ∈ π‘₯ ↔ βˆƒ!𝑦 ∈ ran 𝐹 π‘œ ∈ 𝑦))
7170cbvralvw 3228 . 2 (βˆ€π‘ž ∈ 𝐴 βˆƒ!π‘₯ ∈ ran 𝐹 π‘ž ∈ π‘₯ ↔ βˆ€π‘œ ∈ 𝐴 βˆƒ!𝑦 ∈ ran 𝐹 π‘œ ∈ 𝑦)
7263, 71sylibr 233 1 ((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) β†’ βˆ€π‘ž ∈ 𝐴 βˆƒ!π‘₯ ∈ ran 𝐹 π‘ž ∈ π‘₯)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  βˆƒwrex 3074  βˆƒ!wreu 3354   ∩ cin 3914  {csn 4591  ran crn 5639   Fn wfn 6496  βŸΆwf 6497  β€“1-1β†’wf1 6498  β€˜cfv 6501
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 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fv 6509
This theorem is referenced by:  fvineqsneq  35912
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