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Theorem fvineqsneu 36596
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 7082 . . . . . 6 ((𝐹 Fn 𝐴 ∧ π‘œ ∈ 𝐴) β†’ (πΉβ€˜π‘œ) ∈ ran 𝐹)
21ex 412 . . . . 5 (𝐹 Fn 𝐴 β†’ (π‘œ ∈ 𝐴 β†’ (πΉβ€˜π‘œ) ∈ ran 𝐹))
32adantr 480 . . . 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 6951 . . . . . . . . . 10 (𝐹 Fn 𝐴 β†’ ran 𝐹 = {𝑦 ∣ βˆƒπ‘ ∈ 𝐴 𝑦 = (πΉβ€˜π‘)})
54eqabrd 2875 . . . . . . . . 9 (𝐹 Fn 𝐴 β†’ (𝑦 ∈ ran 𝐹 ↔ βˆƒπ‘ ∈ 𝐴 𝑦 = (πΉβ€˜π‘)))
65adantr 480 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) β†’ (𝑦 ∈ ran 𝐹 ↔ βˆƒπ‘ ∈ 𝐴 𝑦 = (πΉβ€˜π‘)))
7 nfv 1916 . . . . . . . . . 10 Ⅎ𝑝 𝐹 Fn 𝐴
8 nfra1 3280 . . . . . . . . . 10 β„²π‘βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}
97, 8nfan 1901 . . . . . . . . 9 Ⅎ𝑝(𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝})
10 nfv 1916 . . . . . . . . 9 β„²π‘βˆ€π‘œ ∈ 𝐴 (π‘œ ∈ 𝑦 ↔ 𝑦 = (πΉβ€˜π‘œ))
11 eleq2 2821 . . . . . . . . . . . . . . . . . 18 (𝑦 = (πΉβ€˜π‘) β†’ (π‘œ ∈ 𝑦 ↔ π‘œ ∈ (πΉβ€˜π‘)))
12 elin 3964 . . . . . . . . . . . . . . . . . . . . 21 (π‘œ ∈ ((πΉβ€˜π‘) ∩ 𝐴) ↔ (π‘œ ∈ (πΉβ€˜π‘) ∧ π‘œ ∈ 𝐴))
1312rbaib 538 . . . . . . . . . . . . . . . . . . . 20 (π‘œ ∈ 𝐴 β†’ (π‘œ ∈ ((πΉβ€˜π‘) ∩ 𝐴) ↔ π‘œ ∈ (πΉβ€˜π‘)))
1413ad2antll 726 . . . . . . . . . . . . . . . . . . 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 3243 . . . . . . . . . . . . . . . . . . . . . . 23 (βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝} β†’ (𝑝 ∈ 𝐴 β†’ ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}))
16 eleq2 2821 . . . . . . . . . . . . . . . . . . . . . . . 24 (((πΉβ€˜π‘) ∩ 𝐴) = {𝑝} β†’ (π‘œ ∈ ((πΉβ€˜π‘) ∩ 𝐴) ↔ π‘œ ∈ {𝑝}))
17 velsn 4644 . . . . . . . . . . . . . . . . . . . . . . . . 25 (π‘œ ∈ {𝑝} ↔ π‘œ = 𝑝)
18 equcom 2020 . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . 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 491 . . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . . . . 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 510 . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . 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 652 . . . . . . . . . . . . . . 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 454 . . . . . . . . . . . . . 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 649 . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . . 18 (𝑦 = (πΉβ€˜π‘) β†’ (𝑦 = (πΉβ€˜π‘œ) ↔ (πΉβ€˜π‘) = (πΉβ€˜π‘œ)))
32 dffn3 6730 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟢ran 𝐹)
33 fvineqsnf1 36595 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐹:𝐴⟢ran 𝐹 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) β†’ 𝐹:𝐴–1-1β†’ran 𝐹)
3432, 33sylanb 580 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹 Fn 𝐴 ∧ βˆ€π‘ ∈ 𝐴 ((πΉβ€˜π‘) ∩ 𝐴) = {𝑝}) β†’ 𝐹:𝐴–1-1β†’ran 𝐹)
35 dff13 7257 . . . . . . . . . . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . . . . . . . . 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 3243 . . . . . . . . . . . . . . . . . . . . . 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 3243 . . . . . . . . . . . . . . . . . . . . 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 418 . . . . . . . . . . . . . . . . . . 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 6891 . . . . . . . . . . . . . . . . . . 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 510 . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . 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 652 . . . . . . . . . . . . . . 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 454 . . . . . . . . . . . . . 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 649 . . . . . . . . . . . . 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 412 . . . . . . . . . . 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 3144 . . . . . . . . . 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 420 . . . . . . . . 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 3262 . . . . . . . 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 3243 . . . . . . 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 3151 . . . 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 3724 . . . . 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 412 . . . 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 3144 . 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 1916 . . . 4 β„²π‘₯ π‘ž = π‘œ
65 nfv 1916 . . . 4 Ⅎ𝑦 π‘ž = π‘œ
66 nfvd 1917 . . . 4 (π‘ž = π‘œ β†’ Ⅎ𝑦 π‘ž ∈ π‘₯)
67 nfvd 1917 . . . 4 (π‘ž = π‘œ β†’ β„²π‘₯ π‘œ ∈ 𝑦)
68 eleq12 2822 . . . . 5 ((π‘ž = π‘œ ∧ π‘₯ = 𝑦) β†’ (π‘ž ∈ π‘₯ ↔ π‘œ ∈ 𝑦))
6968ex 412 . . . 4 (π‘ž = π‘œ β†’ (π‘₯ = 𝑦 β†’ (π‘ž ∈ π‘₯ ↔ π‘œ ∈ 𝑦)))
7064, 65, 66, 67, 69cbvreud 36558 . . 3 (π‘ž = π‘œ β†’ (βˆƒ!π‘₯ ∈ ran 𝐹 π‘ž ∈ π‘₯ ↔ βˆƒ!𝑦 ∈ ran 𝐹 π‘œ ∈ 𝑦))
7170cbvralvw 3233 . 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 395   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060  βˆƒwrex 3069  βˆƒ!wreu 3373   ∩ cin 3947  {csn 4628  ran crn 5677   Fn wfn 6538  βŸΆwf 6539  β€“1-1β†’wf1 6540  β€˜cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fv 6551
This theorem is referenced by:  fvineqsneq  36597
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