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Theorem disjrnmpt2 39874
Description: Disjointness of the range of a function in map-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypothesis
Ref Expression
disjrnmpt2.1 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
disjrnmpt2 (Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ ran 𝐹 𝑦)
Distinct variable groups:   𝑥,𝐴   𝑦,𝐹
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥,𝑦)   𝐹(𝑥)

Proof of Theorem disjrnmpt2
Dummy variables 𝑢 𝑧 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . . . 7 (𝑦 = 𝑤𝑦 = 𝑤)
21cbvdisjv 4783 . . . . . 6 (Disj 𝑦 ∈ ran 𝐹 𝑦Disj 𝑤 ∈ ran 𝐹 𝑤)
32notbii 309 . . . . 5 Disj 𝑦 ∈ ran 𝐹 𝑦 ↔ ¬ Disj 𝑤 ∈ ran 𝐹 𝑤)
4 id 22 . . . . . . 7 (𝑤 = 𝑣𝑤 = 𝑣)
54ndisj2 39717 . . . . . 6 Disj 𝑤 ∈ ran 𝐹 𝑤 ↔ ∃𝑤 ∈ ran 𝐹𝑣 ∈ ran 𝐹(𝑤𝑣 ∧ (𝑤𝑣) ≠ ∅))
65biimpi 206 . . . . 5 Disj 𝑤 ∈ ran 𝐹 𝑤 → ∃𝑤 ∈ ran 𝐹𝑣 ∈ ran 𝐹(𝑤𝑣 ∧ (𝑤𝑣) ≠ ∅))
73, 6sylbi 207 . . . 4 Disj 𝑦 ∈ ran 𝐹 𝑦 → ∃𝑤 ∈ ran 𝐹𝑣 ∈ ran 𝐹(𝑤𝑣 ∧ (𝑤𝑣) ≠ ∅))
8 disjrnmpt2.1 . . . . . . . . . . . . . 14 𝐹 = (𝑥𝐴𝐵)
98elrnmpt 5527 . . . . . . . . . . . . 13 (𝑤 ∈ ran 𝐹 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝑤 = 𝐵))
109ibi 256 . . . . . . . . . . . 12 (𝑤 ∈ ran 𝐹 → ∃𝑥𝐴 𝑤 = 𝐵)
1110adantr 472 . . . . . . . . . . 11 ((𝑤 ∈ ran 𝐹𝑣 ∈ ran 𝐹) → ∃𝑥𝐴 𝑤 = 𝐵)
12 nfcv 2902 . . . . . . . . . . . . . . . 16 𝑧𝐵
13 nfcsb1v 3690 . . . . . . . . . . . . . . . 16 𝑥𝑧 / 𝑥𝐵
14 csbeq1a 3683 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
1512, 13, 14cbvmpt 4901 . . . . . . . . . . . . . . 15 (𝑥𝐴𝐵) = (𝑧𝐴𝑧 / 𝑥𝐵)
168, 15eqtri 2782 . . . . . . . . . . . . . 14 𝐹 = (𝑧𝐴𝑧 / 𝑥𝐵)
1716elrnmpt 5527 . . . . . . . . . . . . 13 (𝑣 ∈ ran 𝐹 → (𝑣 ∈ ran 𝐹 ↔ ∃𝑧𝐴 𝑣 = 𝑧 / 𝑥𝐵))
1817ibi 256 . . . . . . . . . . . 12 (𝑣 ∈ ran 𝐹 → ∃𝑧𝐴 𝑣 = 𝑧 / 𝑥𝐵)
1918adantl 473 . . . . . . . . . . 11 ((𝑤 ∈ ran 𝐹𝑣 ∈ ran 𝐹) → ∃𝑧𝐴 𝑣 = 𝑧 / 𝑥𝐵)
2011, 19jca 555 . . . . . . . . . 10 ((𝑤 ∈ ran 𝐹𝑣 ∈ ran 𝐹) → (∃𝑥𝐴 𝑤 = 𝐵 ∧ ∃𝑧𝐴 𝑣 = 𝑧 / 𝑥𝐵))
21 nfv 1992 . . . . . . . . . . 11 𝑧 𝑤 = 𝐵
22 nfcv 2902 . . . . . . . . . . . 12 𝑥𝑣
2322, 13nfeq 2914 . . . . . . . . . . 11 𝑥 𝑣 = 𝑧 / 𝑥𝐵
2421, 23reean 3244 . . . . . . . . . 10 (∃𝑥𝐴𝑧𝐴 (𝑤 = 𝐵𝑣 = 𝑧 / 𝑥𝐵) ↔ (∃𝑥𝐴 𝑤 = 𝐵 ∧ ∃𝑧𝐴 𝑣 = 𝑧 / 𝑥𝐵))
2520, 24sylibr 224 . . . . . . . . 9 ((𝑤 ∈ ran 𝐹𝑣 ∈ ran 𝐹) → ∃𝑥𝐴𝑧𝐴 (𝑤 = 𝐵𝑣 = 𝑧 / 𝑥𝐵))
2625adantr 472 . . . . . . . 8 (((𝑤 ∈ ran 𝐹𝑣 ∈ ran 𝐹) ∧ (𝑤𝑣 ∧ (𝑤𝑣) ≠ ∅)) → ∃𝑥𝐴𝑧𝐴 (𝑤 = 𝐵𝑣 = 𝑧 / 𝑥𝐵))
27 nfcv 2902 . . . . . . . . . . . 12 𝑥𝑤
28 nfmpt1 4899 . . . . . . . . . . . . . 14 𝑥(𝑥𝐴𝐵)
298, 28nfcxfr 2900 . . . . . . . . . . . . 13 𝑥𝐹
3029nfrn 5523 . . . . . . . . . . . 12 𝑥ran 𝐹
3127, 30nfel 2915 . . . . . . . . . . 11 𝑥 𝑤 ∈ ran 𝐹
3230nfcri 2896 . . . . . . . . . . 11 𝑥 𝑣 ∈ ran 𝐹
3331, 32nfan 1977 . . . . . . . . . 10 𝑥(𝑤 ∈ ran 𝐹𝑣 ∈ ran 𝐹)
34 nfv 1992 . . . . . . . . . 10 𝑥(𝑤𝑣 ∧ (𝑤𝑣) ≠ ∅)
3533, 34nfan 1977 . . . . . . . . 9 𝑥((𝑤 ∈ ran 𝐹𝑣 ∈ ran 𝐹) ∧ (𝑤𝑣 ∧ (𝑤𝑣) ≠ ∅))
36 simpll 807 . . . . . . . . . . . . . . . . . . 19 (((𝑤 = 𝐵𝑣 = 𝑧 / 𝑥𝐵) ∧ 𝑥 = 𝑧) → 𝑤 = 𝐵)
3714adantl 473 . . . . . . . . . . . . . . . . . . 19 (((𝑤 = 𝐵𝑣 = 𝑧 / 𝑥𝐵) ∧ 𝑥 = 𝑧) → 𝐵 = 𝑧 / 𝑥𝐵)
38 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = 𝑧 / 𝑥𝐵𝑣 = 𝑧 / 𝑥𝐵)
3938eqcomd 2766 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝑧 / 𝑥𝐵𝑧 / 𝑥𝐵 = 𝑣)
4039ad2antlr 765 . . . . . . . . . . . . . . . . . . 19 (((𝑤 = 𝐵𝑣 = 𝑧 / 𝑥𝐵) ∧ 𝑥 = 𝑧) → 𝑧 / 𝑥𝐵 = 𝑣)
4136, 37, 403eqtrd 2798 . . . . . . . . . . . . . . . . . 18 (((𝑤 = 𝐵𝑣 = 𝑧 / 𝑥𝐵) ∧ 𝑥 = 𝑧) → 𝑤 = 𝑣)
4241adantll 752 . . . . . . . . . . . . . . . . 17 (((𝑤𝑣 ∧ (𝑤 = 𝐵𝑣 = 𝑧 / 𝑥𝐵)) ∧ 𝑥 = 𝑧) → 𝑤 = 𝑣)
43 simpll 807 . . . . . . . . . . . . . . . . . 18 (((𝑤𝑣 ∧ (𝑤 = 𝐵𝑣 = 𝑧 / 𝑥𝐵)) ∧ 𝑥 = 𝑧) → 𝑤𝑣)
4443neneqd 2937 . . . . . . . . . . . . . . . . 17 (((𝑤𝑣 ∧ (𝑤 = 𝐵𝑣 = 𝑧 / 𝑥𝐵)) ∧ 𝑥 = 𝑧) → ¬ 𝑤 = 𝑣)
4542, 44pm2.65da 601 . . . . . . . . . . . . . . . 16 ((𝑤𝑣 ∧ (𝑤 = 𝐵𝑣 = 𝑧 / 𝑥𝐵)) → ¬ 𝑥 = 𝑧)
4645neqned 2939 . . . . . . . . . . . . . . 15 ((𝑤𝑣 ∧ (𝑤 = 𝐵𝑣 = 𝑧 / 𝑥𝐵)) → 𝑥𝑧)
4746adantlr 753 . . . . . . . . . . . . . 14 (((𝑤𝑣 ∧ (𝑤𝑣) ≠ ∅) ∧ (𝑤 = 𝐵𝑣 = 𝑧 / 𝑥𝐵)) → 𝑥𝑧)
48 id 22 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝐵𝑤 = 𝐵)
4948eqcomd 2766 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝐵𝐵 = 𝑤)
5049ad2antrl 766 . . . . . . . . . . . . . . . . 17 (((𝑤𝑣) ≠ ∅ ∧ (𝑤 = 𝐵𝑣 = 𝑧 / 𝑥𝐵)) → 𝐵 = 𝑤)
5139ad2antll 767 . . . . . . . . . . . . . . . . 17 (((𝑤𝑣) ≠ ∅ ∧ (𝑤 = 𝐵𝑣 = 𝑧 / 𝑥𝐵)) → 𝑧 / 𝑥𝐵 = 𝑣)
5250, 51ineq12d 3958 . . . . . . . . . . . . . . . 16 (((𝑤𝑣) ≠ ∅ ∧ (𝑤 = 𝐵𝑣 = 𝑧 / 𝑥𝐵)) → (𝐵𝑧 / 𝑥𝐵) = (𝑤𝑣))
53 simpl 474 . . . . . . . . . . . . . . . 16 (((𝑤𝑣) ≠ ∅ ∧ (𝑤 = 𝐵𝑣 = 𝑧 / 𝑥𝐵)) → (𝑤𝑣) ≠ ∅)
5452, 53eqnetrd 2999 . . . . . . . . . . . . . . 15 (((𝑤𝑣) ≠ ∅ ∧ (𝑤 = 𝐵𝑣 = 𝑧 / 𝑥𝐵)) → (𝐵𝑧 / 𝑥𝐵) ≠ ∅)
5554adantll 752 . . . . . . . . . . . . . 14 (((𝑤𝑣 ∧ (𝑤𝑣) ≠ ∅) ∧ (𝑤 = 𝐵𝑣 = 𝑧 / 𝑥𝐵)) → (𝐵𝑧 / 𝑥𝐵) ≠ ∅)
5647, 55jca 555 . . . . . . . . . . . . 13 (((𝑤𝑣 ∧ (𝑤𝑣) ≠ ∅) ∧ (𝑤 = 𝐵𝑣 = 𝑧 / 𝑥𝐵)) → (𝑥𝑧 ∧ (𝐵𝑧 / 𝑥𝐵) ≠ ∅))
5756ex 449 . . . . . . . . . . . 12 ((𝑤𝑣 ∧ (𝑤𝑣) ≠ ∅) → ((𝑤 = 𝐵𝑣 = 𝑧 / 𝑥𝐵) → (𝑥𝑧 ∧ (𝐵𝑧 / 𝑥𝐵) ≠ ∅)))
5857adantl 473 . . . . . . . . . . 11 (((𝑤 ∈ ran 𝐹𝑣 ∈ ran 𝐹) ∧ (𝑤𝑣 ∧ (𝑤𝑣) ≠ ∅)) → ((𝑤 = 𝐵𝑣 = 𝑧 / 𝑥𝐵) → (𝑥𝑧 ∧ (𝐵𝑧 / 𝑥𝐵) ≠ ∅)))
5958reximdv 3154 . . . . . . . . . 10 (((𝑤 ∈ ran 𝐹𝑣 ∈ ran 𝐹) ∧ (𝑤𝑣 ∧ (𝑤𝑣) ≠ ∅)) → (∃𝑧𝐴 (𝑤 = 𝐵𝑣 = 𝑧 / 𝑥𝐵) → ∃𝑧𝐴 (𝑥𝑧 ∧ (𝐵𝑧 / 𝑥𝐵) ≠ ∅)))
6059a1d 25 . . . . . . . . 9 (((𝑤 ∈ ran 𝐹𝑣 ∈ ran 𝐹) ∧ (𝑤𝑣 ∧ (𝑤𝑣) ≠ ∅)) → (𝑥𝐴 → (∃𝑧𝐴 (𝑤 = 𝐵𝑣 = 𝑧 / 𝑥𝐵) → ∃𝑧𝐴 (𝑥𝑧 ∧ (𝐵𝑧 / 𝑥𝐵) ≠ ∅))))
6135, 60reximdai 3150 . . . . . . . 8 (((𝑤 ∈ ran 𝐹𝑣 ∈ ran 𝐹) ∧ (𝑤𝑣 ∧ (𝑤𝑣) ≠ ∅)) → (∃𝑥𝐴𝑧𝐴 (𝑤 = 𝐵𝑣 = 𝑧 / 𝑥𝐵) → ∃𝑥𝐴𝑧𝐴 (𝑥𝑧 ∧ (𝐵𝑧 / 𝑥𝐵) ≠ ∅)))
6226, 61mpd 15 . . . . . . 7 (((𝑤 ∈ ran 𝐹𝑣 ∈ ran 𝐹) ∧ (𝑤𝑣 ∧ (𝑤𝑣) ≠ ∅)) → ∃𝑥𝐴𝑧𝐴 (𝑥𝑧 ∧ (𝐵𝑧 / 𝑥𝐵) ≠ ∅))
6362ex 449 . . . . . 6 ((𝑤 ∈ ran 𝐹𝑣 ∈ ran 𝐹) → ((𝑤𝑣 ∧ (𝑤𝑣) ≠ ∅) → ∃𝑥𝐴𝑧𝐴 (𝑥𝑧 ∧ (𝐵𝑧 / 𝑥𝐵) ≠ ∅)))
6463a1i 11 . . . . 5 Disj 𝑦 ∈ ran 𝐹 𝑦 → ((𝑤 ∈ ran 𝐹𝑣 ∈ ran 𝐹) → ((𝑤𝑣 ∧ (𝑤𝑣) ≠ ∅) → ∃𝑥𝐴𝑧𝐴 (𝑥𝑧 ∧ (𝐵𝑧 / 𝑥𝐵) ≠ ∅))))
6564rexlimdvv 3175 . . . 4 Disj 𝑦 ∈ ran 𝐹 𝑦 → (∃𝑤 ∈ ran 𝐹𝑣 ∈ ran 𝐹(𝑤𝑣 ∧ (𝑤𝑣) ≠ ∅) → ∃𝑥𝐴𝑧𝐴 (𝑥𝑧 ∧ (𝐵𝑧 / 𝑥𝐵) ≠ ∅)))
667, 65mpd 15 . . 3 Disj 𝑦 ∈ ran 𝐹 𝑦 → ∃𝑥𝐴𝑧𝐴 (𝑥𝑧 ∧ (𝐵𝑧 / 𝑥𝐵) ≠ ∅))
67 nfcv 2902 . . . . . 6 𝑢𝐵
68 nfcsb1v 3690 . . . . . 6 𝑥𝑢 / 𝑥𝐵
69 csbeq1a 3683 . . . . . 6 (𝑥 = 𝑢𝐵 = 𝑢 / 𝑥𝐵)
7067, 68, 69cbvdisj 4782 . . . . 5 (Disj 𝑥𝐴 𝐵Disj 𝑢𝐴 𝑢 / 𝑥𝐵)
7170notbii 309 . . . 4 Disj 𝑥𝐴 𝐵 ↔ ¬ Disj 𝑢𝐴 𝑢 / 𝑥𝐵)
72 csbeq1a 3683 . . . . . . 7 (𝑢 = 𝑧𝑢 / 𝑥𝐵 = 𝑧 / 𝑢𝑢 / 𝑥𝐵)
73 csbco 3684 . . . . . . . 8 𝑧 / 𝑢𝑢 / 𝑥𝐵 = 𝑧 / 𝑥𝐵
7473a1i 11 . . . . . . 7 (𝑢 = 𝑧𝑧 / 𝑢𝑢 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)
7572, 74eqtrd 2794 . . . . . 6 (𝑢 = 𝑧𝑢 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)
7675ndisj2 39717 . . . . 5 Disj 𝑢𝐴 𝑢 / 𝑥𝐵 ↔ ∃𝑢𝐴𝑧𝐴 (𝑢𝑧 ∧ (𝑢 / 𝑥𝐵𝑧 / 𝑥𝐵) ≠ ∅))
77 nfcv 2902 . . . . . . 7 𝑥𝐴
78 nfv 1992 . . . . . . . 8 𝑥 𝑢𝑧
7968, 13nfin 3963 . . . . . . . . 9 𝑥(𝑢 / 𝑥𝐵𝑧 / 𝑥𝐵)
80 nfcv 2902 . . . . . . . . 9 𝑥
8179, 80nfne 3032 . . . . . . . 8 𝑥(𝑢 / 𝑥𝐵𝑧 / 𝑥𝐵) ≠ ∅
8278, 81nfan 1977 . . . . . . 7 𝑥(𝑢𝑧 ∧ (𝑢 / 𝑥𝐵𝑧 / 𝑥𝐵) ≠ ∅)
8377, 82nfrex 3145 . . . . . 6 𝑥𝑧𝐴 (𝑢𝑧 ∧ (𝑢 / 𝑥𝐵𝑧 / 𝑥𝐵) ≠ ∅)
84 nfv 1992 . . . . . 6 𝑢𝑧𝐴 (𝑥𝑧 ∧ (𝐵𝑧 / 𝑥𝐵) ≠ ∅)
85 neeq1 2994 . . . . . . . 8 (𝑢 = 𝑥 → (𝑢𝑧𝑥𝑧))
86 csbeq1 3677 . . . . . . . . . . 11 (𝑢 = 𝑥𝑢 / 𝑥𝐵 = 𝑥 / 𝑥𝐵)
87 csbid 3682 . . . . . . . . . . . 12 𝑥 / 𝑥𝐵 = 𝐵
8887a1i 11 . . . . . . . . . . 11 (𝑢 = 𝑥𝑥 / 𝑥𝐵 = 𝐵)
8986, 88eqtrd 2794 . . . . . . . . . 10 (𝑢 = 𝑥𝑢 / 𝑥𝐵 = 𝐵)
9089ineq1d 3956 . . . . . . . . 9 (𝑢 = 𝑥 → (𝑢 / 𝑥𝐵𝑧 / 𝑥𝐵) = (𝐵𝑧 / 𝑥𝐵))
9190neeq1d 2991 . . . . . . . 8 (𝑢 = 𝑥 → ((𝑢 / 𝑥𝐵𝑧 / 𝑥𝐵) ≠ ∅ ↔ (𝐵𝑧 / 𝑥𝐵) ≠ ∅))
9285, 91anbi12d 749 . . . . . . 7 (𝑢 = 𝑥 → ((𝑢𝑧 ∧ (𝑢 / 𝑥𝐵𝑧 / 𝑥𝐵) ≠ ∅) ↔ (𝑥𝑧 ∧ (𝐵𝑧 / 𝑥𝐵) ≠ ∅)))
9392rexbidv 3190 . . . . . 6 (𝑢 = 𝑥 → (∃𝑧𝐴 (𝑢𝑧 ∧ (𝑢 / 𝑥𝐵𝑧 / 𝑥𝐵) ≠ ∅) ↔ ∃𝑧𝐴 (𝑥𝑧 ∧ (𝐵𝑧 / 𝑥𝐵) ≠ ∅)))
9483, 84, 93cbvrex 3307 . . . . 5 (∃𝑢𝐴𝑧𝐴 (𝑢𝑧 ∧ (𝑢 / 𝑥𝐵𝑧 / 𝑥𝐵) ≠ ∅) ↔ ∃𝑥𝐴𝑧𝐴 (𝑥𝑧 ∧ (𝐵𝑧 / 𝑥𝐵) ≠ ∅))
9576, 94bitri 264 . . . 4 Disj 𝑢𝐴 𝑢 / 𝑥𝐵 ↔ ∃𝑥𝐴𝑧𝐴 (𝑥𝑧 ∧ (𝐵𝑧 / 𝑥𝐵) ≠ ∅))
9671, 95bitri 264 . . 3 Disj 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑧𝐴 (𝑥𝑧 ∧ (𝐵𝑧 / 𝑥𝐵) ≠ ∅))
9766, 96sylibr 224 . 2 Disj 𝑦 ∈ ran 𝐹 𝑦 → ¬ Disj 𝑥𝐴 𝐵)
9897con4i 113 1 (Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ ran 𝐹 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1632  wcel 2139  wne 2932  wrex 3051  csb 3674  cin 3714  c0 4058  Disj wdisj 4772  cmpt 4881  ran crn 5267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-disj 4773  df-br 4805  df-opab 4865  df-mpt 4882  df-cnv 5274  df-dm 5276  df-rn 5277
This theorem is referenced by:  meadjiun  41186
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