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Theorem fvineqsneq 34845
 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 4378 . . . . . . . . . . . . . . . . . . . 20 (𝑍 ⊊ ran 𝐹 → ∃𝑥(𝑥 ∈ ran 𝐹 ∧ ¬ 𝑥𝑍))
21adantl 485 . . . . . . . . . . . . . . . . . . 19 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑥(𝑥 ∈ ran 𝐹 ∧ ¬ 𝑥𝑍))
3 df-rex 3112 . . . . . . . . . . . . . . . . . . 19 (∃𝑥 ∈ ran 𝐹 ¬ 𝑥𝑍 ↔ ∃𝑥(𝑥 ∈ ran 𝐹 ∧ ¬ 𝑥𝑍))
42, 3sylibr 237 . . . . . . . . . . . . . . . . . 18 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑥 ∈ ran 𝐹 ¬ 𝑥𝑍)
5 fnrnfv 6700 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑥 ∣ ∃𝑝𝐴 𝑥 = (𝐹𝑝)})
65abeq2d 2924 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹 Fn 𝐴 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑝𝐴 𝑥 = (𝐹𝑝)))
76biimpd 232 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 Fn 𝐴 → (𝑥 ∈ ran 𝐹 → ∃𝑝𝐴 𝑥 = (𝐹𝑝)))
87ralrimiv 3148 . . . . . . . . . . . . . . . . . . . 20 (𝐹 Fn 𝐴 → ∀𝑥 ∈ ran 𝐹𝑝𝐴 𝑥 = (𝐹𝑝))
98adantr 484 . . . . . . . . . . . . . . . . . . 19 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → ∀𝑥 ∈ ran 𝐹𝑝𝐴 𝑥 = (𝐹𝑝))
109adantr 484 . . . . . . . . . . . . . . . . . 18 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∀𝑥 ∈ ran 𝐹𝑝𝐴 𝑥 = (𝐹𝑝))
11 r19.29r 3217 . . . . . . . . . . . . . . . . . 18 ((∃𝑥 ∈ ran 𝐹 ¬ 𝑥𝑍 ∧ ∀𝑥 ∈ ran 𝐹𝑝𝐴 𝑥 = (𝐹𝑝)) → ∃𝑥 ∈ ran 𝐹𝑥𝑍 ∧ ∃𝑝𝐴 𝑥 = (𝐹𝑝)))
124, 10, 11syl2anc 587 . . . . . . . . . . . . . . . . 17 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑥 ∈ ran 𝐹𝑥𝑍 ∧ ∃𝑝𝐴 𝑥 = (𝐹𝑝)))
13 nfra1 3183 . . . . . . . . . . . . . . . . . . . . . 22 𝑝𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}
14 rsp 3170 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} → (𝑝𝐴 → ((𝐹𝑝) ∩ 𝐴) = {𝑝}))
15 vsnid 4562 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝑝 ∈ {𝑝}
16 eleq2 2878 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝐹𝑝) ∩ 𝐴) = {𝑝} → (𝑝 ∈ ((𝐹𝑝) ∩ 𝐴) ↔ 𝑝 ∈ {𝑝}))
1715, 16mpbiri 261 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝐹𝑝) ∩ 𝐴) = {𝑝} → 𝑝 ∈ ((𝐹𝑝) ∩ 𝐴))
1817elin1d 4125 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐹𝑝) ∩ 𝐴) = {𝑝} → 𝑝 ∈ (𝐹𝑝))
1914, 18syl6 35 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} → (𝑝𝐴𝑝 ∈ (𝐹𝑝)))
2019adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} ∧ 𝑥 = (𝐹𝑝)) → (𝑝𝐴𝑝 ∈ (𝐹𝑝)))
21 eleq2 2878 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = (𝐹𝑝) → (𝑝𝑥𝑝 ∈ (𝐹𝑝)))
2221adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} ∧ 𝑥 = (𝐹𝑝)) → (𝑝𝑥𝑝 ∈ (𝐹𝑝)))
2320, 22sylibrd 262 . . . . . . . . . . . . . . . . . . . . . . . 24 ((∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} ∧ 𝑥 = (𝐹𝑝)) → (𝑝𝐴𝑝𝑥))
2423ex 416 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} → (𝑥 = (𝐹𝑝) → (𝑝𝐴𝑝𝑥)))
2524com23 86 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} → (𝑝𝐴 → (𝑥 = (𝐹𝑝) → 𝑝𝑥)))
2613, 25reximdai 3270 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} → (∃𝑝𝐴 𝑥 = (𝐹𝑝) → ∃𝑝𝐴 𝑝𝑥))
2726adantl 485 . . . . . . . . . . . . . . . . . . . 20 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (∃𝑝𝐴 𝑥 = (𝐹𝑝) → ∃𝑝𝐴 𝑝𝑥))
2827adantr 484 . . . . . . . . . . . . . . . . . . 19 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (∃𝑝𝐴 𝑥 = (𝐹𝑝) → ∃𝑝𝐴 𝑝𝑥))
2928anim2d 614 . . . . . . . . . . . . . . . . . 18 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ((¬ 𝑥𝑍 ∧ ∃𝑝𝐴 𝑥 = (𝐹𝑝)) → (¬ 𝑥𝑍 ∧ ∃𝑝𝐴 𝑝𝑥)))
3029reximdv 3232 . . . . . . . . . . . . . . . . 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 464 . . . . . . . . . . . . . . . . . 18 ((¬ 𝑥𝑍 ∧ ∃𝑝𝐴 𝑝𝑥) ↔ (∃𝑝𝐴 𝑝𝑥 ∧ ¬ 𝑥𝑍))
33 r19.41v 3300 . . . . . . . . . . . . . . . . . 18 (∃𝑝𝐴 (𝑝𝑥 ∧ ¬ 𝑥𝑍) ↔ (∃𝑝𝐴 𝑝𝑥 ∧ ¬ 𝑥𝑍))
3432, 33bitr4i 281 . . . . . . . . . . . . . . . . 17 ((¬ 𝑥𝑍 ∧ ∃𝑝𝐴 𝑝𝑥) ↔ ∃𝑝𝐴 (𝑝𝑥 ∧ ¬ 𝑥𝑍))
3534rexbii 3210 . . . . . . . . . . . . . . . 16 (∃𝑥 ∈ ran 𝐹𝑥𝑍 ∧ ∃𝑝𝐴 𝑝𝑥) ↔ ∃𝑥 ∈ ran 𝐹𝑝𝐴 (𝑝𝑥 ∧ ¬ 𝑥𝑍))
3631, 35sylib 221 . . . . . . . . . . . . . . 15 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑥 ∈ ran 𝐹𝑝𝐴 (𝑝𝑥 ∧ ¬ 𝑥𝑍))
37 rexcom 3308 . . . . . . . . . . . . . . 15 (∃𝑝𝐴𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) ↔ ∃𝑥 ∈ ran 𝐹𝑝𝐴 (𝑝𝑥 ∧ ¬ 𝑥𝑍))
3836, 37sylibr 237 . . . . . . . . . . . . . 14 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑝𝐴𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍))
39 nfre1 3265 . . . . . . . . . . . . . . . . 17 𝑥𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍)
403919.3 2200 . . . . . . . . . . . . . . . 16 (∀𝑥𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) ↔ ∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍))
41 alral 3122 . . . . . . . . . . . . . . . 16 (∀𝑥𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) → ∀𝑥 ∈ ran 𝐹𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍))
4240, 41sylbir 238 . . . . . . . . . . . . . . 15 (∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) → ∀𝑥 ∈ ran 𝐹𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍))
4342reximi 3206 . . . . . . . . . . . . . 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 1915 . . . . . . . . . . . . . . . 16 𝑝 𝐹 Fn 𝐴
4645, 13nfan 1900 . . . . . . . . . . . . . . 15 𝑝(𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝})
47 nfv 1915 . . . . . . . . . . . . . . 15 𝑝 𝑍 ⊊ ran 𝐹
4846, 47nfan 1900 . . . . . . . . . . . . . 14 𝑝((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹)
49 nfv 1915 . . . . . . . . . . . . . . . 16 𝑥(((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ 𝑝𝐴)
50 fvineqsneu 34844 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → ∀𝑝𝐴 ∃!𝑥 ∈ ran 𝐹 𝑝𝑥)
5150adantr 484 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∀𝑝𝐴 ∃!𝑥 ∈ ran 𝐹 𝑝𝑥)
52 rsp 3170 . . . . . . . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . . . . . 19 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ((𝑝𝐴𝑥 ∈ ran 𝐹) → ∃!𝑥 ∈ ran 𝐹 𝑝𝑥))
5554imp 410 . . . . . . . . . . . . . . . . . 18 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ (𝑝𝐴𝑥 ∈ ran 𝐹)) → ∃!𝑥 ∈ ran 𝐹 𝑝𝑥)
56 reupick3 4240 . . . . . . . . . . . . . . . . . . . . . 22 ((∃!𝑥 ∈ ran 𝐹 𝑝𝑥 ∧ ∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) ∧ 𝑥 ∈ ran 𝐹) → (𝑝𝑥 → ¬ 𝑥𝑍))
57563expa 1115 . . . . . . . . . . . . . . . . . . . . 21 (((∃!𝑥 ∈ ran 𝐹 𝑝𝑥 ∧ ∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍)) ∧ 𝑥 ∈ ran 𝐹) → (𝑝𝑥 → ¬ 𝑥𝑍))
5857expcom 417 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ran 𝐹 → ((∃!𝑥 ∈ ran 𝐹 𝑝𝑥 ∧ ∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍)) → (𝑝𝑥 → ¬ 𝑥𝑍)))
5958adantl 485 . . . . . . . . . . . . . . . . . . 19 ((𝑝𝐴𝑥 ∈ ran 𝐹) → ((∃!𝑥 ∈ ran 𝐹 𝑝𝑥 ∧ ∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍)) → (𝑝𝑥 → ¬ 𝑥𝑍)))
6059adantl 485 . . . . . . . . . . . . . . . . . 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 694 . . . . . . . . . . . . . . . . 17 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ (𝑝𝐴𝑥 ∈ ran 𝐹)) → (∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) → (𝑝𝑥 → ¬ 𝑥𝑍)))
6261expr 460 . . . . . . . . . . . . . . . 16 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ 𝑝𝐴) → (𝑥 ∈ ran 𝐹 → (∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) → (𝑝𝑥 → ¬ 𝑥𝑍))))
6349, 62ralrimi 3180 . . . . . . . . . . . . . . 15 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ 𝑝𝐴) → ∀𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) → (𝑝𝑥 → ¬ 𝑥𝑍)))
6463ex 416 . . . . . . . . . . . . . 14 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (𝑝𝐴 → ∀𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) → (𝑝𝑥 → ¬ 𝑥𝑍))))
6548, 64ralrimi 3180 . . . . . . . . . . . . 13 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∀𝑝𝐴𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) → (𝑝𝑥 → ¬ 𝑥𝑍)))
66 r19.29r 3217 . . . . . . . . . . . . 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 587 . . . . . . . . . . . 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 3130 . . . . . . . . . . . . . 14 (∀𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) → (𝑝𝑥 → ¬ 𝑥𝑍)) → (∀𝑥 ∈ ran 𝐹𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) → ∀𝑥 ∈ ran 𝐹(𝑝𝑥 → ¬ 𝑥𝑍)))
6968impcom 411 . . . . . . . . . . . . 13 ((∀𝑥 ∈ ran 𝐹𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) ∧ ∀𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝𝑥 ∧ ¬ 𝑥𝑍) → (𝑝𝑥 → ¬ 𝑥𝑍))) → ∀𝑥 ∈ ran 𝐹(𝑝𝑥 → ¬ 𝑥𝑍))
7069reximi 3206 . . . . . . . . . . . 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 363 . . . . . . . . . . . . . . 15 ((𝑝𝑥 → ¬ 𝑥𝑍) ↔ (𝑥𝑍 → ¬ 𝑝𝑥))
7372ralbii 3133 . . . . . . . . . . . . . 14 (∀𝑥 ∈ ran 𝐹(𝑝𝑥 → ¬ 𝑥𝑍) ↔ ∀𝑥 ∈ ran 𝐹(𝑥𝑍 → ¬ 𝑝𝑥))
74 df-ral 3111 . . . . . . . . . . . . . 14 (∀𝑥 ∈ ran 𝐹(𝑥𝑍 → ¬ 𝑝𝑥) ↔ ∀𝑥(𝑥 ∈ ran 𝐹 → (𝑥𝑍 → ¬ 𝑝𝑥)))
75 bi2.04 392 . . . . . . . . . . . . . . 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 300 . . . . . . . . . . . . 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 3279 . . . . . . . . . . 11 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (∃𝑝𝐴𝑥 ∈ ran 𝐹(𝑝𝑥 → ¬ 𝑥𝑍) ↔ ∃𝑝𝐴𝑥(𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥))))
8071, 79mpbid 235 . . . . . . . . . 10 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑝𝐴𝑥(𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥)))
81 nfv 1915 . . . . . . . . . . . . . . 15 𝑥((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹)
82 nfa1 2152 . . . . . . . . . . . . . . 15 𝑥𝑥(𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥))
8381, 82nfan 1900 . . . . . . . . . . . . . 14 𝑥(((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ ∀𝑥(𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥)))
84 pssss 4023 . . . . . . . . . . . . . . . . . . . 20 (𝑍 ⊊ ran 𝐹𝑍 ⊆ ran 𝐹)
85 dfss2 3901 . . . . . . . . . . . . . . . . . . . 20 (𝑍 ⊆ ran 𝐹 ↔ ∀𝑥(𝑥𝑍𝑥 ∈ ran 𝐹))
8684, 85sylib 221 . . . . . . . . . . . . . . . . . . 19 (𝑍 ⊊ ran 𝐹 → ∀𝑥(𝑥𝑍𝑥 ∈ ran 𝐹))
8786adantl 485 . . . . . . . . . . . . . . . . . 18 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∀𝑥(𝑥𝑍𝑥 ∈ ran 𝐹))
88 df-ral 3111 . . . . . . . . . . . . . . . . . 18 (∀𝑥𝑍 𝑥 ∈ ran 𝐹 ↔ ∀𝑥(𝑥𝑍𝑥 ∈ ran 𝐹))
8987, 88sylibr 237 . . . . . . . . . . . . . . . . 17 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∀𝑥𝑍 𝑥 ∈ ran 𝐹)
9089adantr 484 . . . . . . . . . . . . . . . 16 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ ∀𝑥(𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥))) → ∀𝑥𝑍 𝑥 ∈ ran 𝐹)
91 rsp 3170 . . . . . . . . . . . . . . . 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 3111 . . . . . . . . . . . . . . . . . 18 (∀𝑥𝑍 (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥) ↔ ∀𝑥(𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥)))
9493biimpri 231 . . . . . . . . . . . . . . . . 17 (∀𝑥(𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥)) → ∀𝑥𝑍 (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥))
9594adantl 485 . . . . . . . . . . . . . . . 16 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ ∀𝑥(𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥))) → ∀𝑥𝑍 (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥))
96 rsp 3170 . . . . . . . . . . . . . . . 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 3180 . . . . . . . . . . . . 13 ((((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ ∀𝑥(𝑥𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝𝑥))) → ∀𝑥𝑍 ¬ 𝑝𝑥)
10099ex 416 . . . . . . . . . . . 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 3270 . . . . . . . . . 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 3199 . . . . . . . . . 10 (∀𝑥𝑍 ¬ 𝑝𝑥 ↔ ¬ ∃𝑥𝑍 𝑝𝑥)
105104rexbii 3210 . . . . . . . . 9 (∃𝑝𝐴𝑥𝑍 ¬ 𝑝𝑥 ↔ ∃𝑝𝐴 ¬ ∃𝑥𝑍 𝑝𝑥)
106103, 105sylib 221 . . . . . . . 8 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑝𝐴 ¬ ∃𝑥𝑍 𝑝𝑥)
107 eluni2 4804 . . . . . . . . . 10 (𝑝 𝑍 ↔ ∃𝑥𝑍 𝑝𝑥)
108107notbii 323 . . . . . . . . 9 𝑝 𝑍 ↔ ¬ ∃𝑥𝑍 𝑝𝑥)
109108rexbii 3210 . . . . . . . 8 (∃𝑝𝐴 ¬ 𝑝 𝑍 ↔ ∃𝑝𝐴 ¬ ∃𝑥𝑍 𝑝𝑥)
110106, 109sylibr 237 . . . . . . 7 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑝𝐴 ¬ 𝑝 𝑍)
111 dfss3 3903 . . . . . . . . 9 (𝐴 𝑍 ↔ ∀𝑝𝐴 𝑝 𝑍)
112 dfral2 3200 . . . . . . . . 9 (∀𝑝𝐴 𝑝 𝑍 ↔ ¬ ∃𝑝𝐴 ¬ 𝑝 𝑍)
113111, 112bitri 278 . . . . . . . 8 (𝐴 𝑍 ↔ ¬ ∃𝑝𝐴 ¬ 𝑝 𝑍)
114113con2bii2 34766 . . . . . . 7 𝐴 𝑍 ↔ ∃𝑝𝐴 ¬ 𝑝 𝑍)
115110, 114sylibr 237 . . . . . 6 (((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ¬ 𝐴 𝑍)
116115ex 416 . . . . 5 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (𝑍 ⊊ ran 𝐹 → ¬ 𝐴 𝑍))
117116con2d 136 . . . 4 ((𝐹 Fn 𝐴 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (𝐴 𝑍 → ¬ 𝑍 ⊊ ran 𝐹))
118 npss 4038 . . . 4 𝑍 ⊊ ran 𝐹 ↔ (𝑍 ⊆ ran 𝐹𝑍 = ran 𝐹))
119117, 118syl6ib 254 . . 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 422 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 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  ∃!wreu 3108   ∩ cin 3880   ⊆ wss 3881   ⊊ wpss 3882  {csn 4525  ∪ cuni 4800  ran crn 5520   Fn wfn 6319  ‘cfv 6324 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 5167  ax-nul 5174  ax-pr 5295 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-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fv 6332 This theorem is referenced by:  pibt2  34850
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