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Theorem fvineqsneq 36288
Description: A theorem about functions where the image of every point intersects the domain only at that point. (Contributed by ML, 28-Mar-2021.)
Assertion
Ref Expression
fvineqsneq (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑍 ⊆ ran 𝐹𝐴 𝑍)) → 𝑍 = ran 𝐹)
Distinct variable groups:   𝐴,𝑝   𝐹,𝑝   𝑍,𝑝

Proof of Theorem fvineqsneq
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pssnel 4470 . . . . . . . . . . . . . . . . . . . 20 (𝑍 ⊊ ran 𝐹 → ∃𝑥(𝑥 ∈ ran 𝐹 ∧ ¬ 𝑥𝑍))
21adantl 482 . . . . . . . . . . . . . . . . . . 19 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑥(𝑥 ∈ ran 𝐹 ∧ ¬ 𝑥𝑍))
3 df-rex 3071 . . . . . . . . . . . . . . . . . . 19 (∃𝑥 ∈ ran 𝐹 ¬ 𝑥𝑍 ↔ ∃𝑥(𝑥 ∈ ran 𝐹 ∧ ¬ 𝑥𝑍))
42, 3sylibr 233 . . . . . . . . . . . . . . . . . 18 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑥 ∈ ran 𝐹 ¬ 𝑥𝑍)
5 fnrnfv 6951 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑥 ∣ ∃𝑝𝐴 𝑥 = (𝐹𝑝)})
65eqabrd 2876 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹 Fn 𝐴 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑝𝐴 𝑥 = (𝐹𝑝)))
76biimpd 228 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 Fn 𝐴 → (𝑥 ∈ ran 𝐹 → ∃𝑝𝐴 𝑥 = (𝐹𝑝)))
87ralrimiv 3145 . . . . . . . . . . . . . . . . . . . 20 (𝐹 Fn 𝐴 → ∀𝑥 ∈ ran 𝐹𝑝𝐴 𝑥 = (𝐹𝑝))
98adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → ∀𝑥 ∈ ran 𝐹𝑝𝐴 𝑥 = (𝐹𝑝))
109adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∀𝑥 ∈ ran 𝐹𝑝𝐴 𝑥 = (𝐹𝑝))
11 r19.29r 3116 . . . . . . . . . . . . . . . . . 18 ((∃𝑥 ∈ ran 𝐹 ¬ 𝑥𝑍 ∧ ∀𝑥 ∈ ran 𝐹𝑝𝐴 𝑥 = (𝐹𝑝)) → ∃𝑥 ∈ ran 𝐹𝑥𝑍 ∧ ∃𝑝𝐴 𝑥 = (𝐹𝑝)))
124, 10, 11syl2anc 584 . . . . . . . . . . . . . . . . 17 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑥 ∈ ran 𝐹𝑥𝑍 ∧ ∃𝑝𝐴 𝑥 = (𝐹𝑝)))
13 nfra1 3281 . . . . . . . . . . . . . . . . . . . . . 22 𝑝𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}
14 rsp 3244 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} → (𝑝𝐴 → ((𝐹𝑝) ∩ 𝐴) = {𝑝}))
15 vsnid 4665 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝑝 ∈ {𝑝}
16 eleq2 2822 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝐹𝑝) ∩ 𝐴) = {𝑝} → (𝑝 ∈ ((𝐹𝑝) ∩ 𝐴) ↔ 𝑝 ∈ {𝑝}))
1715, 16mpbiri 257 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝐹𝑝) ∩ 𝐴) = {𝑝} → 𝑝 ∈ ((𝐹𝑝) ∩ 𝐴))
1817elin1d 4198 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐹𝑝) ∩ 𝐴) = {𝑝} → 𝑝 ∈ (𝐹𝑝))
1914, 18syl6 35 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} → (𝑝𝐴𝑝 ∈ (𝐹𝑝)))
2019adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} ∧ 𝑥 = (𝐹𝑝)) → (𝑝𝐴𝑝 ∈ (𝐹𝑝)))
21 eleq2 2822 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = (𝐹𝑝) → (𝑝𝑥𝑝 ∈ (𝐹𝑝)))
2221adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} ∧ 𝑥 = (𝐹𝑝)) → (𝑝𝑥𝑝 ∈ (𝐹𝑝)))
2320, 22sylibrd 258 . . . . . . . . . . . . . . . . . . . . . . . 24 ((∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} ∧ 𝑥 = (𝐹𝑝)) → (𝑝𝐴𝑝𝑥))
2423ex 413 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} → (𝑥 = (𝐹𝑝) → (𝑝𝐴𝑝𝑥)))
2524com23 86 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} → (𝑝𝐴 → (𝑥 = (𝐹𝑝) → 𝑝𝑥)))
2613, 25reximdai 3258 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} → (∃𝑝𝐴 𝑥 = (𝐹𝑝) → ∃𝑝𝐴 𝑝𝑥))
2726adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (∃𝑝𝐴 𝑥 = (𝐹𝑝) → ∃𝑝𝐴 𝑝𝑥))
2827adantr 481 . . . . . . . . . . . . . . . . . . 19 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (∃𝑝𝐴 𝑥 = (𝐹𝑝) → ∃𝑝𝐴 𝑝𝑥))
2928anim2d 612 . . . . . . . . . . . . . . . . . 18 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ((¬ 𝑥𝑍 ∧ ∃𝑝𝐴 𝑥 = (𝐹𝑝)) → (¬ 𝑥𝑍 ∧ ∃𝑝𝐴 𝑝𝑥)))
3029reximdv 3170 . . . . . . . . . . . . . . . . 17 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (∃𝑥 ∈ ran 𝐹𝑥𝑍 ∧ ∃𝑝𝐴 𝑥 = (𝐹𝑝)) → ∃𝑥 ∈ ran 𝐹𝑥𝑍 ∧ ∃𝑝𝐴 𝑝𝑥)))
3112, 30mpd 15 . . . . . . . . . . . . . . . 16 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑥 ∈ ran 𝐹𝑥𝑍 ∧ ∃𝑝𝐴 𝑝𝑥))
32 ancom 461 . . . . . . . . . . . . . . . . . 18 ((¬ 𝑥𝑍 ∧ ∃𝑝𝐴 𝑝𝑥) ↔ (∃𝑝𝐴 𝑝𝑥 ∧ ¬ 𝑥𝑍))
33 r19.41v 3188 . . . . . . . . . . . . . . . . . 18 (∃𝑝𝐴 (𝑝𝑥 ∧ ¬ 𝑥𝑍) ↔ (∃𝑝𝐴 𝑝𝑥 ∧ ¬ 𝑥𝑍))
3432, 33bitr4i 277 . . . . . . . . . . . . . . . . 17 ((¬ 𝑥𝑍 ∧ ∃𝑝𝐴 𝑝𝑥) ↔ ∃𝑝𝐴 (𝑝𝑥 ∧ ¬ 𝑥𝑍))
3534rexbii 3094 . . . . . . . . . . . . . . . 16 (∃𝑥 ∈ ran 𝐹𝑥𝑍 ∧ ∃𝑝𝐴 𝑝𝑥) ↔ ∃𝑥 ∈ ran 𝐹𝑝𝐴 (𝑝𝑥 ∧ ¬ 𝑥𝑍))
3631, 35sylib 217 . . . . . . . . . . . . . . 15 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑥 ∈ ran 𝐹𝑝𝐴 (𝑝𝑥 ∧ ¬ 𝑥𝑍))
37 rexcom 3287 . . . . . . . . . . . . . . 15 (∃𝑝𝐴𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) ↔ ∃𝑥 ∈ ran 𝐹𝑝𝐴 (𝑝𝑥 ∧ ¬ 𝑥𝑍))
3836, 37sylibr 233 . . . . . . . . . . . . . 14 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑝𝐴𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍))
39 nfre1 3282 . . . . . . . . . . . . . . . . 17 𝑥𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍)
403919.3 2195 . . . . . . . . . . . . . . . 16 (∀𝑥𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) ↔ ∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍))
41 alral 3075 . . . . . . . . . . . . . . . 16 (∀𝑥𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) → ∀𝑥 ∈ ran 𝐹𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍))
4240, 41sylbir 234 . . . . . . . . . . . . . . 15 (∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) → ∀𝑥 ∈ ran 𝐹𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍))
4342reximi 3084 . . . . . . . . . . . . . 14 (∃𝑝𝐴𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) → ∃𝑝𝐴𝑥 ∈ ran 𝐹𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍))
4438, 43syl 17 . . . . . . . . . . . . 13 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑝𝐴𝑥 ∈ ran 𝐹𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍))
45 nfv 1917 . . . . . . . . . . . . . . . 16 𝑝 𝐹 Fn 𝐴
4645, 13nfan 1902 . . . . . . . . . . . . . . 15 𝑝(𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝})
47 nfv 1917 . . . . . . . . . . . . . . 15 𝑝 𝑍 ⊊ ran 𝐹
4846, 47nfan 1902 . . . . . . . . . . . . . 14 𝑝((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹)
49 nfv 1917 . . . . . . . . . . . . . . . 16 𝑥(((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ 𝑝𝐴)
50 fvineqsneu 36287 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → ∀𝑝𝐴 ∃!𝑥 ∈ ran 𝐹 𝑝𝑥)
5150adantr 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∀𝑝𝐴 ∃!𝑥 ∈ ran 𝐹 𝑝𝑥)
52 rsp 3244 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑝𝐴 ∃!𝑥 ∈ ran 𝐹 𝑝𝑥 → (𝑝𝐴 → ∃!𝑥 ∈ ran 𝐹 𝑝𝑥))
5351, 52syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (𝑝𝐴 → ∃!𝑥 ∈ ran 𝐹 𝑝𝑥))
5453adantrd 492 . . . . . . . . . . . . . . . . . . 19 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ((𝑝𝐴𝑥 ∈ ran 𝐹) → ∃!𝑥 ∈ ran 𝐹 𝑝𝑥))
5554imp 407 . . . . . . . . . . . . . . . . . 18 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ (𝑝𝐴𝑥 ∈ ran 𝐹)) → ∃!𝑥 ∈ ran 𝐹 𝑝𝑥)
56 reupick3 4319 . . . . . . . . . . . . . . . . . . . . . 22 ((∃!𝑥 ∈ ran 𝐹 𝑝𝑥 ∧ ∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) ∧ 𝑥 ∈ ran 𝐹) → (𝑝𝑥 → ¬ 𝑥𝑍))
57563expa 1118 . . . . . . . . . . . . . . . . . . . . 21 (((∃!𝑥 ∈ ran 𝐹 𝑝𝑥 ∧ ∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍)) ∧ 𝑥 ∈ ran 𝐹) → (𝑝𝑥 → ¬ 𝑥𝑍))
5857expcom 414 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ran 𝐹 → ((∃!𝑥 ∈ ran 𝐹 𝑝𝑥 ∧ ∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍)) → (𝑝𝑥 → ¬ 𝑥𝑍)))
5958adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝑝𝐴𝑥 ∈ ran 𝐹) → ((∃!𝑥 ∈ ran 𝐹 𝑝𝑥 ∧ ∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍)) → (𝑝𝑥 → ¬ 𝑥𝑍)))
6059adantl 482 . . . . . . . . . . . . . . . . . 18 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ (𝑝𝐴𝑥 ∈ ran 𝐹)) → ((∃!𝑥 ∈ ran 𝐹 𝑝𝑥 ∧ ∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍)) → (𝑝𝑥 → ¬ 𝑥𝑍)))
6155, 60mpand 693 . . . . . . . . . . . . . . . . 17 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ (𝑝𝐴𝑥 ∈ ran 𝐹)) → (∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) → (𝑝𝑥 → ¬ 𝑥𝑍)))
6261expr 457 . . . . . . . . . . . . . . . 16 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ 𝑝𝐴) → (𝑥 ∈ ran 𝐹 → (∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) → (𝑝𝑥 → ¬ 𝑥𝑍))))
6349, 62ralrimi 3254 . . . . . . . . . . . . . . 15 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ 𝑝𝐴) → ∀𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) → (𝑝𝑥 → ¬ 𝑥𝑍)))
6463ex 413 . . . . . . . . . . . . . 14 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (𝑝𝐴 → ∀𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) → (𝑝𝑥 → ¬ 𝑥𝑍))))
6548, 64ralrimi 3254 . . . . . . . . . . . . 13 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∀𝑝𝐴𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) → (𝑝𝑥 → ¬ 𝑥𝑍)))
66 r19.29r 3116 . . . . . . . . . . . . 13 ((∃𝑝𝐴𝑥 ∈ ran 𝐹𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) ∧ ∀𝑝𝐴𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) → (𝑝𝑥 → ¬ 𝑥𝑍))) → ∃𝑝𝐴 (∀𝑥 ∈ ran 𝐹𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) ∧ ∀𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) → (𝑝𝑥 → ¬ 𝑥𝑍))))
6744, 65, 66syl2anc 584 . . . . . . . . . . . 12 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑝𝐴 (∀𝑥 ∈ ran 𝐹𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) ∧ ∀𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) → (𝑝𝑥 → ¬ 𝑥𝑍))))
68 ralim 3086 . . . . . . . . . . . . . 14 (∀𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) → (𝑝𝑥 → ¬ 𝑥𝑍)) → (∀𝑥 ∈ ran 𝐹𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) → ∀𝑥 ∈ ran 𝐹(𝑝𝑥 → ¬ 𝑥𝑍)))
6968impcom 408 . . . . . . . . . . . . 13 ((∀𝑥 ∈ ran 𝐹𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) ∧ ∀𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) → (𝑝𝑥 → ¬ 𝑥𝑍))) → ∀𝑥 ∈ ran 𝐹(𝑝𝑥 → ¬ 𝑥𝑍))
7069reximi 3084 . . . . . . . . . . . 12 (∃𝑝𝐴 (∀𝑥 ∈ ran 𝐹𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) ∧ ∀𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) → (𝑝𝑥 → ¬ 𝑥𝑍))) → ∃𝑝𝐴𝑥 ∈ ran 𝐹(𝑝𝑥 → ¬ 𝑥𝑍))
7167, 70syl 17 . . . . . . . . . . 11 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑝𝐴𝑥 ∈ ran 𝐹(𝑝𝑥 → ¬ 𝑥𝑍))
72 con2b 359 . . . . . . . . . . . . . . 15 ((𝑝𝑥 → ¬ 𝑥𝑍) ↔ (𝑥𝑍 → ¬ 𝑝𝑥))
7372ralbii 3093 . . . . . . . . . . . . . 14 (∀𝑥 ∈ ran 𝐹(𝑝𝑥 → ¬ 𝑥𝑍) ↔ ∀𝑥 ∈ ran 𝐹(𝑥𝑍 → ¬ 𝑝𝑥))
74 df-ral 3062 . . . . . . . . . . . . . 14 (∀𝑥 ∈ ran 𝐹(𝑥𝑍 → ¬ 𝑝𝑥) ↔ ∀𝑥(𝑥 ∈ ran 𝐹 → (𝑥𝑍 → ¬ 𝑝𝑥)))
75 bi2.04 388 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ran 𝐹 → (𝑥𝑍 → ¬ 𝑝𝑥)) ↔ (𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥)))
7675albii 1821 . . . . . . . . . . . . . 14 (∀𝑥(𝑥 ∈ ran 𝐹 → (𝑥𝑍 → ¬ 𝑝𝑥)) ↔ ∀𝑥(𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥)))
7773, 74, 763bitri 296 . . . . . . . . . . . . 13 (∀𝑥 ∈ ran 𝐹(𝑝𝑥 → ¬ 𝑥𝑍) ↔ ∀𝑥(𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥)))
7877a1i 11 . . . . . . . . . . . 12 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (∀𝑥 ∈ ran 𝐹(𝑝𝑥 → ¬ 𝑥𝑍) ↔ ∀𝑥(𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥))))
7948, 78rexbid 3271 . . . . . . . . . . 11 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (∃𝑝𝐴𝑥 ∈ ran 𝐹(𝑝𝑥 → ¬ 𝑥𝑍) ↔ ∃𝑝𝐴𝑥(𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥))))
8071, 79mpbid 231 . . . . . . . . . 10 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑝𝐴𝑥(𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥)))
81 nfv 1917 . . . . . . . . . . . . . . 15 𝑥((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹)
82 nfa1 2148 . . . . . . . . . . . . . . 15 𝑥𝑥(𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥))
8381, 82nfan 1902 . . . . . . . . . . . . . 14 𝑥(((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ ∀𝑥(𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥)))
84 pssss 4095 . . . . . . . . . . . . . . . . . . . 20 (𝑍 ⊊ ran 𝐹𝑍 ⊆ ran 𝐹)
85 dfss2 3968 . . . . . . . . . . . . . . . . . . . 20 (𝑍 ⊆ ran 𝐹 ↔ ∀𝑥(𝑥𝑍𝑥 ∈ ran 𝐹))
8684, 85sylib 217 . . . . . . . . . . . . . . . . . . 19 (𝑍 ⊊ ran 𝐹 → ∀𝑥(𝑥𝑍𝑥 ∈ ran 𝐹))
8786adantl 482 . . . . . . . . . . . . . . . . . 18 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∀𝑥(𝑥𝑍𝑥 ∈ ran 𝐹))
88 df-ral 3062 . . . . . . . . . . . . . . . . . 18 (∀𝑥𝑍 𝑥 ∈ ran 𝐹 ↔ ∀𝑥(𝑥𝑍𝑥 ∈ ran 𝐹))
8987, 88sylibr 233 . . . . . . . . . . . . . . . . 17 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∀𝑥𝑍 𝑥 ∈ ran 𝐹)
9089adantr 481 . . . . . . . . . . . . . . . 16 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ ∀𝑥(𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥))) → ∀𝑥𝑍 𝑥 ∈ ran 𝐹)
91 rsp 3244 . . . . . . . . . . . . . . . 16 (∀𝑥𝑍 𝑥 ∈ ran 𝐹 → (𝑥𝑍𝑥 ∈ ran 𝐹))
9290, 91syl 17 . . . . . . . . . . . . . . 15 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ ∀𝑥(𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥))) → (𝑥𝑍𝑥 ∈ ran 𝐹))
93 df-ral 3062 . . . . . . . . . . . . . . . . . 18 (∀𝑥𝑍 (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥) ↔ ∀𝑥(𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥)))
9493biimpri 227 . . . . . . . . . . . . . . . . 17 (∀𝑥(𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥)) → ∀𝑥𝑍 (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥))
9594adantl 482 . . . . . . . . . . . . . . . 16 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ ∀𝑥(𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥))) → ∀𝑥𝑍 (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥))
96 rsp 3244 . . . . . . . . . . . . . . . 16 (∀𝑥𝑍 (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥) → (𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥)))
9795, 96syl 17 . . . . . . . . . . . . . . 15 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ ∀𝑥(𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥))) → (𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥)))
9892, 97mpdd 43 . . . . . . . . . . . . . 14 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ ∀𝑥(𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥))) → (𝑥𝑍 → ¬ 𝑝𝑥))
9983, 98ralrimi 3254 . . . . . . . . . . . . 13 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ ∀𝑥(𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥))) → ∀𝑥𝑍 ¬ 𝑝𝑥)
10099ex 413 . . . . . . . . . . . 12 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (∀𝑥(𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥)) → ∀𝑥𝑍 ¬ 𝑝𝑥))
101100a1d 25 . . . . . . . . . . 11 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (𝑝𝐴 → (∀𝑥(𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥)) → ∀𝑥𝑍 ¬ 𝑝𝑥)))
10248, 101reximdai 3258 . . . . . . . . . 10 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (∃𝑝𝐴𝑥(𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥)) → ∃𝑝𝐴𝑥𝑍 ¬ 𝑝𝑥))
10380, 102mpd 15 . . . . . . . . 9 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑝𝐴𝑥𝑍 ¬ 𝑝𝑥)
104 ralnex 3072 . . . . . . . . . 10 (∀𝑥𝑍 ¬ 𝑝𝑥 ↔ ¬ ∃𝑥𝑍 𝑝𝑥)
105104rexbii 3094 . . . . . . . . 9 (∃𝑝𝐴𝑥𝑍 ¬ 𝑝𝑥 ↔ ∃𝑝𝐴 ¬ ∃𝑥𝑍 𝑝𝑥)
106103, 105sylib 217 . . . . . . . 8 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑝𝐴 ¬ ∃𝑥𝑍 𝑝𝑥)
107 eluni2 4912 . . . . . . . . . 10 (𝑝 𝑍 ↔ ∃𝑥𝑍 𝑝𝑥)
108107notbii 319 . . . . . . . . 9 𝑝 𝑍 ↔ ¬ ∃𝑥𝑍 𝑝𝑥)
109108rexbii 3094 . . . . . . . 8 (∃𝑝𝐴 ¬ 𝑝 𝑍 ↔ ∃𝑝𝐴 ¬ ∃𝑥𝑍 𝑝𝑥)
110106, 109sylibr 233 . . . . . . 7 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑝𝐴 ¬ 𝑝 𝑍)
111 dfss3 3970 . . . . . . . . 9 (𝐴 𝑍 ↔ ∀𝑝𝐴 𝑝 𝑍)
112 dfral2 3099 . . . . . . . . 9 (∀𝑝𝐴 𝑝 𝑍 ↔ ¬ ∃𝑝𝐴 ¬ 𝑝 𝑍)
113111, 112bitri 274 . . . . . . . 8 (𝐴 𝑍 ↔ ¬ ∃𝑝𝐴 ¬ 𝑝 𝑍)
114113con2bii2 36209 . . . . . . 7 𝐴 𝑍 ↔ ∃𝑝𝐴 ¬ 𝑝 𝑍)
115110, 114sylibr 233 . . . . . 6 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ¬ 𝐴 𝑍)
116115ex 413 . . . . 5 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (𝑍 ⊊ ran 𝐹 → ¬ 𝐴 𝑍))
117116con2d 134 . . . 4 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (𝐴 𝑍 → ¬ 𝑍 ⊊ ran 𝐹))
118 npss 4110 . . . 4 𝑍 ⊊ ran 𝐹 ↔ (𝑍 ⊆ ran 𝐹𝑍 = ran 𝐹))
119117, 118imbitrdi 250 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (𝐴 𝑍 → (𝑍 ⊆ ran 𝐹𝑍 = ran 𝐹)))
120119com23 86 . 2 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (𝑍 ⊆ ran 𝐹 → (𝐴 𝑍𝑍 = ran 𝐹)))
121120imp32 419 1 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑍 ⊆ ran 𝐹𝐴 𝑍)) → 𝑍 = ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wal 1539   = wceq 1541  wex 1781  wcel 2106  wral 3061  wrex 3070  ∃!wreu 3374  cin 3947  wss 3948  wpss 3949  {csn 4628   cuni 4908  ran crn 5677   Fn wfn 6538  cfv 6543
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-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  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-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  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:  pibt2  36293
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