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Theorem disjinfi 43402
Description: Only a finite number of disjoint sets can have a nonempty intersection with a finite set 𝐶. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
disjinfi.b ((𝜑𝑥𝐴) → 𝐵𝑉)
disjinfi.d (𝜑Disj 𝑥𝐴 𝐵)
disjinfi.c (𝜑𝐶 ∈ Fin)
Assertion
Ref Expression
disjinfi (𝜑 → {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅} ∈ Fin)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝑉   𝜑,𝑥
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem disjinfi
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 disjinfi.c . . 3 (𝜑𝐶 ∈ Fin)
2 inss2 4189 . . 3 ( ran (𝑥𝐴𝐵) ∩ 𝐶) ⊆ 𝐶
3 ssfi 9117 . . 3 ((𝐶 ∈ Fin ∧ ( ran (𝑥𝐴𝐵) ∩ 𝐶) ⊆ 𝐶) → ( ran (𝑥𝐴𝐵) ∩ 𝐶) ∈ Fin)
41, 2, 3sylancl 586 . 2 (𝜑 → ( ran (𝑥𝐴𝐵) ∩ 𝐶) ∈ Fin)
52a1i 11 . . . 4 (𝜑 → ( ran (𝑥𝐴𝐵) ∩ 𝐶) ⊆ 𝐶)
61, 5ssexd 5281 . . 3 (𝜑 → ( ran (𝑥𝐴𝐵) ∩ 𝐶) ∈ V)
7 elinel1 4155 . . . . . . . . . 10 (𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶) → 𝑦 ran (𝑥𝐴𝐵))
8 eluni2 4869 . . . . . . . . . . . 12 (𝑦 ran (𝑥𝐴𝐵) ↔ ∃𝑤 ∈ ran (𝑥𝐴𝐵)𝑦𝑤)
98biimpi 215 . . . . . . . . . . 11 (𝑦 ran (𝑥𝐴𝐵) → ∃𝑤 ∈ ran (𝑥𝐴𝐵)𝑦𝑤)
10 eqid 2736 . . . . . . . . . . . . . . . . . . 19 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
1110elrnmpt 5911 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ V → (𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑤 = 𝐵))
1211elv 3451 . . . . . . . . . . . . . . . . 17 (𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑤 = 𝐵)
1312biimpi 215 . . . . . . . . . . . . . . . 16 (𝑤 ∈ ran (𝑥𝐴𝐵) → ∃𝑥𝐴 𝑤 = 𝐵)
1413adantr 481 . . . . . . . . . . . . . . 15 ((𝑤 ∈ ran (𝑥𝐴𝐵) ∧ 𝑦𝑤) → ∃𝑥𝐴 𝑤 = 𝐵)
15 nfmpt1 5213 . . . . . . . . . . . . . . . . . . 19 𝑥(𝑥𝐴𝐵)
1615nfrn 5907 . . . . . . . . . . . . . . . . . 18 𝑥ran (𝑥𝐴𝐵)
1716nfcri 2894 . . . . . . . . . . . . . . . . 17 𝑥 𝑤 ∈ ran (𝑥𝐴𝐵)
18 nfv 1917 . . . . . . . . . . . . . . . . 17 𝑥 𝑦𝑤
1917, 18nfan 1902 . . . . . . . . . . . . . . . 16 𝑥(𝑤 ∈ ran (𝑥𝐴𝐵) ∧ 𝑦𝑤)
20 simpl 483 . . . . . . . . . . . . . . . . . . . 20 ((𝑦𝑤𝑤 = 𝐵) → 𝑦𝑤)
21 simpr 485 . . . . . . . . . . . . . . . . . . . 20 ((𝑦𝑤𝑤 = 𝐵) → 𝑤 = 𝐵)
2220, 21eleqtrd 2840 . . . . . . . . . . . . . . . . . . 19 ((𝑦𝑤𝑤 = 𝐵) → 𝑦𝐵)
2322ex 413 . . . . . . . . . . . . . . . . . 18 (𝑦𝑤 → (𝑤 = 𝐵𝑦𝐵))
2423a1d 25 . . . . . . . . . . . . . . . . 17 (𝑦𝑤 → (𝑥𝐴 → (𝑤 = 𝐵𝑦𝐵)))
2524adantl 482 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ ran (𝑥𝐴𝐵) ∧ 𝑦𝑤) → (𝑥𝐴 → (𝑤 = 𝐵𝑦𝐵)))
2619, 25reximdai 3244 . . . . . . . . . . . . . . 15 ((𝑤 ∈ ran (𝑥𝐴𝐵) ∧ 𝑦𝑤) → (∃𝑥𝐴 𝑤 = 𝐵 → ∃𝑥𝐴 𝑦𝐵))
2714, 26mpd 15 . . . . . . . . . . . . . 14 ((𝑤 ∈ ran (𝑥𝐴𝐵) ∧ 𝑦𝑤) → ∃𝑥𝐴 𝑦𝐵)
2827ex 413 . . . . . . . . . . . . 13 (𝑤 ∈ ran (𝑥𝐴𝐵) → (𝑦𝑤 → ∃𝑥𝐴 𝑦𝐵))
2928a1i 11 . . . . . . . . . . . 12 (𝑦 ran (𝑥𝐴𝐵) → (𝑤 ∈ ran (𝑥𝐴𝐵) → (𝑦𝑤 → ∃𝑥𝐴 𝑦𝐵)))
3029rexlimdv 3150 . . . . . . . . . . 11 (𝑦 ran (𝑥𝐴𝐵) → (∃𝑤 ∈ ran (𝑥𝐴𝐵)𝑦𝑤 → ∃𝑥𝐴 𝑦𝐵))
319, 30mpd 15 . . . . . . . . . 10 (𝑦 ran (𝑥𝐴𝐵) → ∃𝑥𝐴 𝑦𝐵)
327, 31syl 17 . . . . . . . . 9 (𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶) → ∃𝑥𝐴 𝑦𝐵)
3332adantl 482 . . . . . . . 8 ((𝜑𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶)) → ∃𝑥𝐴 𝑦𝐵)
34 nfv 1917 . . . . . . . . . 10 𝑥𝜑
3516nfuni 4872 . . . . . . . . . . . 12 𝑥 ran (𝑥𝐴𝐵)
36 nfcv 2907 . . . . . . . . . . . 12 𝑥𝐶
3735, 36nfin 4176 . . . . . . . . . . 11 𝑥( ran (𝑥𝐴𝐵) ∩ 𝐶)
3837nfcri 2894 . . . . . . . . . 10 𝑥 𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶)
3934, 38nfan 1902 . . . . . . . . 9 𝑥(𝜑𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶))
40 nfre1 3268 . . . . . . . . 9 𝑥𝑥𝐴 𝑦 ∈ (𝐵𝐶)
41 elinel2 4156 . . . . . . . . . . 11 (𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶) → 𝑦𝐶)
42 simp2 1137 . . . . . . . . . . . . 13 ((𝑦𝐶𝑥𝐴𝑦𝐵) → 𝑥𝐴)
43 simpr 485 . . . . . . . . . . . . . 14 ((𝑦𝐶𝑦𝐵) → 𝑦𝐵)
44 simpl 483 . . . . . . . . . . . . . 14 ((𝑦𝐶𝑦𝐵) → 𝑦𝐶)
4543, 44elind 4154 . . . . . . . . . . . . 13 ((𝑦𝐶𝑦𝐵) → 𝑦 ∈ (𝐵𝐶))
46 rspe 3232 . . . . . . . . . . . . 13 ((𝑥𝐴𝑦 ∈ (𝐵𝐶)) → ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
4742, 45, 463imp3i2an 1345 . . . . . . . . . . . 12 ((𝑦𝐶𝑥𝐴𝑦𝐵) → ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
48473exp 1119 . . . . . . . . . . 11 (𝑦𝐶 → (𝑥𝐴 → (𝑦𝐵 → ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))))
4941, 48syl 17 . . . . . . . . . 10 (𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶) → (𝑥𝐴 → (𝑦𝐵 → ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))))
5049adantl 482 . . . . . . . . 9 ((𝜑𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶)) → (𝑥𝐴 → (𝑦𝐵 → ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))))
5139, 40, 50rexlimd 3249 . . . . . . . 8 ((𝜑𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶)) → (∃𝑥𝐴 𝑦𝐵 → ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶)))
5233, 51mpd 15 . . . . . . 7 ((𝜑𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶)) → ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
53 disjinfi.d . . . . . . . . . . . . . . 15 (𝜑Disj 𝑥𝐴 𝐵)
54 disjors 5086 . . . . . . . . . . . . . . 15 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑧𝐴𝑤𝐴 (𝑧 = 𝑤 ∨ (𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵) = ∅))
5553, 54sylib 217 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑧𝐴𝑤𝐴 (𝑧 = 𝑤 ∨ (𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵) = ∅))
56 nfv 1917 . . . . . . . . . . . . . . 15 𝑧𝑤𝐴 (𝑥 = 𝑤 ∨ (𝐵𝑤 / 𝑥𝐵) = ∅)
57 nfcv 2907 . . . . . . . . . . . . . . . 16 𝑥𝐴
58 nfv 1917 . . . . . . . . . . . . . . . . 17 𝑥 𝑧 = 𝑤
59 nfcsb1v 3880 . . . . . . . . . . . . . . . . . . 19 𝑥𝑧 / 𝑥𝐵
60 nfcv 2907 . . . . . . . . . . . . . . . . . . . 20 𝑥𝑤
6160nfcsb1 3879 . . . . . . . . . . . . . . . . . . 19 𝑥𝑤 / 𝑥𝐵
6259, 61nfin 4176 . . . . . . . . . . . . . . . . . 18 𝑥(𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵)
6362nfeq1 2922 . . . . . . . . . . . . . . . . 17 𝑥(𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵) = ∅
6458, 63nfor 1907 . . . . . . . . . . . . . . . 16 𝑥(𝑧 = 𝑤 ∨ (𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵) = ∅)
6557, 64nfralw 3294 . . . . . . . . . . . . . . 15 𝑥𝑤𝐴 (𝑧 = 𝑤 ∨ (𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵) = ∅)
66 equequ1 2028 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (𝑥 = 𝑤𝑧 = 𝑤))
67 csbeq1a 3869 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
6867ineq1d 4171 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑧 → (𝐵𝑤 / 𝑥𝐵) = (𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵))
6968eqeq1d 2738 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → ((𝐵𝑤 / 𝑥𝐵) = ∅ ↔ (𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵) = ∅))
7066, 69orbi12d 917 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → ((𝑥 = 𝑤 ∨ (𝐵𝑤 / 𝑥𝐵) = ∅) ↔ (𝑧 = 𝑤 ∨ (𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵) = ∅)))
7170ralbidv 3174 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (∀𝑤𝐴 (𝑥 = 𝑤 ∨ (𝐵𝑤 / 𝑥𝐵) = ∅) ↔ ∀𝑤𝐴 (𝑧 = 𝑤 ∨ (𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵) = ∅)))
7256, 65, 71cbvralw 3289 . . . . . . . . . . . . . 14 (∀𝑥𝐴𝑤𝐴 (𝑥 = 𝑤 ∨ (𝐵𝑤 / 𝑥𝐵) = ∅) ↔ ∀𝑧𝐴𝑤𝐴 (𝑧 = 𝑤 ∨ (𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵) = ∅))
7355, 72sylibr 233 . . . . . . . . . . . . 13 (𝜑 → ∀𝑥𝐴𝑤𝐴 (𝑥 = 𝑤 ∨ (𝐵𝑤 / 𝑥𝐵) = ∅))
7473r19.21bi 3234 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ∀𝑤𝐴 (𝑥 = 𝑤 ∨ (𝐵𝑤 / 𝑥𝐵) = ∅))
75 rspa 3231 . . . . . . . . . . . . 13 ((∀𝑤𝐴 (𝑥 = 𝑤 ∨ (𝐵𝑤 / 𝑥𝐵) = ∅) ∧ 𝑤𝐴) → (𝑥 = 𝑤 ∨ (𝐵𝑤 / 𝑥𝐵) = ∅))
7675orcomd 869 . . . . . . . . . . . 12 ((∀𝑤𝐴 (𝑥 = 𝑤 ∨ (𝐵𝑤 / 𝑥𝐵) = ∅) ∧ 𝑤𝐴) → ((𝐵𝑤 / 𝑥𝐵) = ∅ ∨ 𝑥 = 𝑤))
7774, 76sylan 580 . . . . . . . . . . 11 (((𝜑𝑥𝐴) ∧ 𝑤𝐴) → ((𝐵𝑤 / 𝑥𝐵) = ∅ ∨ 𝑥 = 𝑤))
78 elinel1 4155 . . . . . . . . . . . 12 (𝑦 ∈ (𝐵𝐶) → 𝑦𝐵)
79 sbsbc 3743 . . . . . . . . . . . . . 14 ([𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶) ↔ [𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶))
80 sbcel2 4375 . . . . . . . . . . . . . 14 ([𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶) ↔ 𝑦𝑤 / 𝑥(𝐵𝐶))
81 csbin 4399 . . . . . . . . . . . . . . 15 𝑤 / 𝑥(𝐵𝐶) = (𝑤 / 𝑥𝐵𝑤 / 𝑥𝐶)
8281eleq2i 2829 . . . . . . . . . . . . . 14 (𝑦𝑤 / 𝑥(𝐵𝐶) ↔ 𝑦 ∈ (𝑤 / 𝑥𝐵𝑤 / 𝑥𝐶))
8379, 80, 823bitri 296 . . . . . . . . . . . . 13 ([𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶) ↔ 𝑦 ∈ (𝑤 / 𝑥𝐵𝑤 / 𝑥𝐶))
84 elinel1 4155 . . . . . . . . . . . . 13 (𝑦 ∈ (𝑤 / 𝑥𝐵𝑤 / 𝑥𝐶) → 𝑦𝑤 / 𝑥𝐵)
8583, 84sylbi 216 . . . . . . . . . . . 12 ([𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶) → 𝑦𝑤 / 𝑥𝐵)
86 inelcm 4424 . . . . . . . . . . . . 13 ((𝑦𝐵𝑦𝑤 / 𝑥𝐵) → (𝐵𝑤 / 𝑥𝐵) ≠ ∅)
8786neneqd 2948 . . . . . . . . . . . 12 ((𝑦𝐵𝑦𝑤 / 𝑥𝐵) → ¬ (𝐵𝑤 / 𝑥𝐵) = ∅)
8878, 85, 87syl2an 596 . . . . . . . . . . 11 ((𝑦 ∈ (𝐵𝐶) ∧ [𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶)) → ¬ (𝐵𝑤 / 𝑥𝐵) = ∅)
89 pm2.53 849 . . . . . . . . . . 11 (((𝐵𝑤 / 𝑥𝐵) = ∅ ∨ 𝑥 = 𝑤) → (¬ (𝐵𝑤 / 𝑥𝐵) = ∅ → 𝑥 = 𝑤))
9077, 88, 89syl2im 40 . . . . . . . . . 10 (((𝜑𝑥𝐴) ∧ 𝑤𝐴) → ((𝑦 ∈ (𝐵𝐶) ∧ [𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶)) → 𝑥 = 𝑤))
9190ralrimiva 3143 . . . . . . . . 9 ((𝜑𝑥𝐴) → ∀𝑤𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ [𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶)) → 𝑥 = 𝑤))
9291ralrimiva 3143 . . . . . . . 8 (𝜑 → ∀𝑥𝐴𝑤𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ [𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶)) → 𝑥 = 𝑤))
9392adantr 481 . . . . . . 7 ((𝜑𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶)) → ∀𝑥𝐴𝑤𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ [𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶)) → 𝑥 = 𝑤))
94 reu2 3683 . . . . . . 7 (∃!𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ∧ ∀𝑥𝐴𝑤𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ [𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶)) → 𝑥 = 𝑤)))
9552, 93, 94sylanbrc 583 . . . . . 6 ((𝜑𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶)) → ∃!𝑥𝐴 𝑦 ∈ (𝐵𝐶))
96 riotacl2 7330 . . . . . 6 (∃!𝑥𝐴 𝑦 ∈ (𝐵𝐶) → (𝑥𝐴 𝑦 ∈ (𝐵𝐶)) ∈ {𝑥𝐴𝑦 ∈ (𝐵𝐶)})
97 nfriota1 7320 . . . . . . . . 9 𝑥(𝑥𝐴 𝑦 ∈ (𝐵𝐶))
9897nfcsb1 3879 . . . . . . . . . . 11 𝑥(𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵
9998, 36nfin 4176 . . . . . . . . . 10 𝑥((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵𝐶)
10099nfcri 2894 . . . . . . . . 9 𝑥 𝑦 ∈ ((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵𝐶)
101 csbeq1a 3869 . . . . . . . . . . 11 (𝑥 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶)) → 𝐵 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵)
102101ineq1d 4171 . . . . . . . . . 10 (𝑥 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶)) → (𝐵𝐶) = ((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵𝐶))
103102eleq2d 2823 . . . . . . . . 9 (𝑥 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶)) → (𝑦 ∈ (𝐵𝐶) ↔ 𝑦 ∈ ((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵𝐶)))
10497, 57, 100, 103elrabf 3641 . . . . . . . 8 ((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) ∈ {𝑥𝐴𝑦 ∈ (𝐵𝐶)} ↔ ((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) ∈ 𝐴𝑦 ∈ ((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵𝐶)))
105104simplbi 498 . . . . . . 7 ((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) ∈ {𝑥𝐴𝑦 ∈ (𝐵𝐶)} → (𝑥𝐴 𝑦 ∈ (𝐵𝐶)) ∈ 𝐴)
106104simprbi 497 . . . . . . . 8 ((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) ∈ {𝑥𝐴𝑦 ∈ (𝐵𝐶)} → 𝑦 ∈ ((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵𝐶))
107106ne0d 4295 . . . . . . 7 ((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) ∈ {𝑥𝐴𝑦 ∈ (𝐵𝐶)} → ((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵𝐶) ≠ ∅)
108 nfcv 2907 . . . . . . . . 9 𝑥
10999, 108nfne 3045 . . . . . . . 8 𝑥((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵𝐶) ≠ ∅
110102neeq1d 3003 . . . . . . . 8 (𝑥 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶)) → ((𝐵𝐶) ≠ ∅ ↔ ((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵𝐶) ≠ ∅))
11197, 57, 109, 110elrabf 3641 . . . . . . 7 ((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) ∈ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅} ↔ ((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) ∈ 𝐴 ∧ ((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵𝐶) ≠ ∅))
112105, 107, 111sylanbrc 583 . . . . . 6 ((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) ∈ {𝑥𝐴𝑦 ∈ (𝐵𝐶)} → (𝑥𝐴 𝑦 ∈ (𝐵𝐶)) ∈ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅})
11395, 96, 1123syl 18 . . . . 5 ((𝜑𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶)) → (𝑥𝐴 𝑦 ∈ (𝐵𝐶)) ∈ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅})
114113ralrimiva 3143 . . . 4 (𝜑 → ∀𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶)(𝑥𝐴 𝑦 ∈ (𝐵𝐶)) ∈ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅})
11561, 36nfin 4176 . . . . . . . . . . . 12 𝑥(𝑤 / 𝑥𝐵𝐶)
116115, 108nfne 3045 . . . . . . . . . . 11 𝑥(𝑤 / 𝑥𝐵𝐶) ≠ ∅
117 csbeq1a 3869 . . . . . . . . . . . . 13 (𝑥 = 𝑤𝐵 = 𝑤 / 𝑥𝐵)
118117ineq1d 4171 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (𝐵𝐶) = (𝑤 / 𝑥𝐵𝐶))
119118neeq1d 3003 . . . . . . . . . . 11 (𝑥 = 𝑤 → ((𝐵𝐶) ≠ ∅ ↔ (𝑤 / 𝑥𝐵𝐶) ≠ ∅))
12060, 57, 116, 119elrabf 3641 . . . . . . . . . 10 (𝑤 ∈ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅} ↔ (𝑤𝐴 ∧ (𝑤 / 𝑥𝐵𝐶) ≠ ∅))
121120simprbi 497 . . . . . . . . 9 (𝑤 ∈ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅} → (𝑤 / 𝑥𝐵𝐶) ≠ ∅)
122 n0 4306 . . . . . . . . 9 ((𝑤 / 𝑥𝐵𝐶) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶))
123121, 122sylib 217 . . . . . . . 8 (𝑤 ∈ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅} → ∃𝑦 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶))
124123adantl 482 . . . . . . 7 ((𝜑𝑤 ∈ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅}) → ∃𝑦 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶))
125120simplbi 498 . . . . . . . . 9 (𝑤 ∈ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅} → 𝑤𝐴)
126 elinel1 4155 . . . . . . . . . . . . . 14 (𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) → 𝑦𝑤 / 𝑥𝐵)
127126adantl 482 . . . . . . . . . . . . 13 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑦𝑤 / 𝑥𝐵)
128 simplr 767 . . . . . . . . . . . . . . 15 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑤𝐴)
129 nfv 1917 . . . . . . . . . . . . . . . . . 18 𝑥(𝜑𝑤𝐴)
13061nfel1 2923 . . . . . . . . . . . . . . . . . 18 𝑥𝑤 / 𝑥𝐵𝑉
131129, 130nfim 1899 . . . . . . . . . . . . . . . . 17 𝑥((𝜑𝑤𝐴) → 𝑤 / 𝑥𝐵𝑉)
132 eleq1w 2820 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑤 → (𝑥𝐴𝑤𝐴))
133132anbi2d 629 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑤 → ((𝜑𝑥𝐴) ↔ (𝜑𝑤𝐴)))
134117eleq1d 2822 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑤 → (𝐵𝑉𝑤 / 𝑥𝐵𝑉))
135133, 134imbi12d 344 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑤 → (((𝜑𝑥𝐴) → 𝐵𝑉) ↔ ((𝜑𝑤𝐴) → 𝑤 / 𝑥𝐵𝑉)))
136 disjinfi.b . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → 𝐵𝑉)
137131, 135, 136chvarfv 2233 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝐴) → 𝑤 / 𝑥𝐵𝑉)
138137adantr 481 . . . . . . . . . . . . . . 15 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑤 / 𝑥𝐵𝑉)
139 eqid 2736 . . . . . . . . . . . . . . . 16 (𝑤𝐴𝑤 / 𝑥𝐵) = (𝑤𝐴𝑤 / 𝑥𝐵)
140139elrnmpt1 5913 . . . . . . . . . . . . . . 15 ((𝑤𝐴𝑤 / 𝑥𝐵𝑉) → 𝑤 / 𝑥𝐵 ∈ ran (𝑤𝐴𝑤 / 𝑥𝐵))
141128, 138, 140syl2anc 584 . . . . . . . . . . . . . 14 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑤 / 𝑥𝐵 ∈ ran (𝑤𝐴𝑤 / 𝑥𝐵))
142 nfcv 2907 . . . . . . . . . . . . . . . 16 𝑤𝐵
143117equcoms 2023 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑥𝐵 = 𝑤 / 𝑥𝐵)
144143eqcomd 2742 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑥𝑤 / 𝑥𝐵 = 𝐵)
14561, 142, 144cbvmpt 5216 . . . . . . . . . . . . . . 15 (𝑤𝐴𝑤 / 𝑥𝐵) = (𝑥𝐴𝐵)
146145rneqi 5892 . . . . . . . . . . . . . 14 ran (𝑤𝐴𝑤 / 𝑥𝐵) = ran (𝑥𝐴𝐵)
147141, 146eleqtrdi 2848 . . . . . . . . . . . . 13 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑤 / 𝑥𝐵 ∈ ran (𝑥𝐴𝐵))
148 elunii 4870 . . . . . . . . . . . . 13 ((𝑦𝑤 / 𝑥𝐵𝑤 / 𝑥𝐵 ∈ ran (𝑥𝐴𝐵)) → 𝑦 ran (𝑥𝐴𝐵))
149127, 147, 148syl2anc 584 . . . . . . . . . . . 12 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑦 ran (𝑥𝐴𝐵))
150 elinel2 4156 . . . . . . . . . . . . 13 (𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) → 𝑦𝐶)
151150adantl 482 . . . . . . . . . . . 12 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑦𝐶)
152149, 151elind 4154 . . . . . . . . . . 11 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶))
153 nfv 1917 . . . . . . . . . . . . 13 𝑤 𝑦 ∈ (𝐵𝐶)
154115nfcri 2894 . . . . . . . . . . . . 13 𝑥 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)
155118eleq2d 2823 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (𝑦 ∈ (𝐵𝐶) ↔ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)))
156153, 154, 155cbvriotaw 7322 . . . . . . . . . . . 12 (𝑥𝐴 𝑦 ∈ (𝐵𝐶)) = (𝑤𝐴 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶))
157 simpr 485 . . . . . . . . . . . . 13 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶))
158 rspe 3232 . . . . . . . . . . . . . . . 16 ((𝑤𝐴𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → ∃𝑤𝐴 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶))
159158adantll 712 . . . . . . . . . . . . . . 15 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → ∃𝑤𝐴 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶))
160 simpll 765 . . . . . . . . . . . . . . . 16 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝜑)
161 sbequ 2086 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑧 → ([𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶) ↔ [𝑧 / 𝑥]𝑦 ∈ (𝐵𝐶)))
162 sbsbc 3743 . . . . . . . . . . . . . . . . . . . . . . . 24 ([𝑧 / 𝑥]𝑦 ∈ (𝐵𝐶) ↔ [𝑧 / 𝑥]𝑦 ∈ (𝐵𝐶))
163162a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑧 → ([𝑧 / 𝑥]𝑦 ∈ (𝐵𝐶) ↔ [𝑧 / 𝑥]𝑦 ∈ (𝐵𝐶)))
164 sbcel2 4375 . . . . . . . . . . . . . . . . . . . . . . . . 25 ([𝑧 / 𝑥]𝑦 ∈ (𝐵𝐶) ↔ 𝑦𝑧 / 𝑥(𝐵𝐶))
165 csbin 4399 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑧 / 𝑥(𝐵𝐶) = (𝑧 / 𝑥𝐵𝑧 / 𝑥𝐶)
166 csbconstg 3874 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 ∈ V → 𝑧 / 𝑥𝐶 = 𝐶)
167166elv 3451 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑧 / 𝑥𝐶 = 𝐶
168167ineq2i 4169 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 / 𝑥𝐵𝑧 / 𝑥𝐶) = (𝑧 / 𝑥𝐵𝐶)
169165, 168eqtri 2764 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑧 / 𝑥(𝐵𝐶) = (𝑧 / 𝑥𝐵𝐶)
170169eleq2i 2829 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦𝑧 / 𝑥(𝐵𝐶) ↔ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶))
171164, 170bitri 274 . . . . . . . . . . . . . . . . . . . . . . . 24 ([𝑧 / 𝑥]𝑦 ∈ (𝐵𝐶) ↔ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶))
172171a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑧 → ([𝑧 / 𝑥]𝑦 ∈ (𝐵𝐶) ↔ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)))
173161, 163, 1723bitrd 304 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = 𝑧 → ([𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶) ↔ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)))
174173anbi2d 629 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑧 → ((𝑦 ∈ (𝐵𝐶) ∧ [𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶)) ↔ (𝑦 ∈ (𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶))))
175 equequ2 2029 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑧 → (𝑥 = 𝑤𝑥 = 𝑧))
176174, 175imbi12d 344 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑧 → (((𝑦 ∈ (𝐵𝐶) ∧ [𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶)) → 𝑥 = 𝑤) ↔ ((𝑦 ∈ (𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑥 = 𝑧)))
177176cbvralvw 3225 . . . . . . . . . . . . . . . . . . 19 (∀𝑤𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ [𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶)) → 𝑥 = 𝑤) ↔ ∀𝑧𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑥 = 𝑧))
178177ralbii 3096 . . . . . . . . . . . . . . . . . 18 (∀𝑥𝐴𝑤𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ [𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶)) → 𝑥 = 𝑤) ↔ ∀𝑥𝐴𝑧𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑥 = 𝑧))
179 nfv 1917 . . . . . . . . . . . . . . . . . . 19 𝑤𝑧𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑥 = 𝑧)
18059, 36nfin 4176 . . . . . . . . . . . . . . . . . . . . . . 23 𝑥(𝑧 / 𝑥𝐵𝐶)
181180nfcri 2894 . . . . . . . . . . . . . . . . . . . . . 22 𝑥 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)
182154, 181nfan 1902 . . . . . . . . . . . . . . . . . . . . 21 𝑥(𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶))
183 nfv 1917 . . . . . . . . . . . . . . . . . . . . 21 𝑥 𝑤 = 𝑧
184182, 183nfim 1899 . . . . . . . . . . . . . . . . . . . 20 𝑥((𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑤 = 𝑧)
18557, 184nfralw 3294 . . . . . . . . . . . . . . . . . . 19 𝑥𝑧𝐴 ((𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑤 = 𝑧)
186155anbi1d 630 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑤 → ((𝑦 ∈ (𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) ↔ (𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶))))
187 equequ1 2028 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑤 → (𝑥 = 𝑧𝑤 = 𝑧))
188186, 187imbi12d 344 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑤 → (((𝑦 ∈ (𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑥 = 𝑧) ↔ ((𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑤 = 𝑧)))
189188ralbidv 3174 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑤 → (∀𝑧𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑥 = 𝑧) ↔ ∀𝑧𝐴 ((𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑤 = 𝑧)))
190179, 185, 189cbvralw 3289 . . . . . . . . . . . . . . . . . 18 (∀𝑥𝐴𝑧𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑥 = 𝑧) ↔ ∀𝑤𝐴𝑧𝐴 ((𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑤 = 𝑧))
191 sbsbc 3743 . . . . . . . . . . . . . . . . . . . . . 22 ([𝑧 / 𝑤]𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ↔ [𝑧 / 𝑤]𝑦 ∈ (𝑤 / 𝑥𝐵𝐶))
192 sbcel2 4375 . . . . . . . . . . . . . . . . . . . . . 22 ([𝑧 / 𝑤]𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ↔ 𝑦𝑧 / 𝑤(𝑤 / 𝑥𝐵𝐶))
193 csbin 4399 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑧 / 𝑤(𝑤 / 𝑥𝐵𝐶) = (𝑧 / 𝑤𝑤 / 𝑥𝐵𝑧 / 𝑤𝐶)
194 csbcow 3870 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑧 / 𝑤𝑤 / 𝑥𝐵 = 𝑧 / 𝑥𝐵
195 csbconstg 3874 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 ∈ V → 𝑧 / 𝑤𝐶 = 𝐶)
196195elv 3451 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑧 / 𝑤𝐶 = 𝐶
197194, 196ineq12i 4170 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 / 𝑤𝑤 / 𝑥𝐵𝑧 / 𝑤𝐶) = (𝑧 / 𝑥𝐵𝐶)
198193, 197eqtri 2764 . . . . . . . . . . . . . . . . . . . . . . 23 𝑧 / 𝑤(𝑤 / 𝑥𝐵𝐶) = (𝑧 / 𝑥𝐵𝐶)
199198eleq2i 2829 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦𝑧 / 𝑤(𝑤 / 𝑥𝐵𝐶) ↔ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶))
200191, 192, 1993bitrri 297 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (𝑧 / 𝑥𝐵𝐶) ↔ [𝑧 / 𝑤]𝑦 ∈ (𝑤 / 𝑥𝐵𝐶))
201200anbi2i 623 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) ↔ (𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ∧ [𝑧 / 𝑤]𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)))
202201imbi1i 349 . . . . . . . . . . . . . . . . . . 19 (((𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑤 = 𝑧) ↔ ((𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ∧ [𝑧 / 𝑤]𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑤 = 𝑧))
2032022ralbii 3127 . . . . . . . . . . . . . . . . . 18 (∀𝑤𝐴𝑧𝐴 ((𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑤 = 𝑧) ↔ ∀𝑤𝐴𝑧𝐴 ((𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ∧ [𝑧 / 𝑤]𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑤 = 𝑧))
204178, 190, 2033bitri 296 . . . . . . . . . . . . . . . . 17 (∀𝑥𝐴𝑤𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ [𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶)) → 𝑥 = 𝑤) ↔ ∀𝑤𝐴𝑧𝐴 ((𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ∧ [𝑧 / 𝑤]𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑤 = 𝑧))
20593, 204sylib 217 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶)) → ∀𝑤𝐴𝑧𝐴 ((𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ∧ [𝑧 / 𝑤]𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑤 = 𝑧))
206160, 152, 205syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → ∀𝑤𝐴𝑧𝐴 ((𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ∧ [𝑧 / 𝑤]𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑤 = 𝑧))
207 reu2 3683 . . . . . . . . . . . . . . 15 (∃!𝑤𝐴 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ↔ (∃𝑤𝐴 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ∧ ∀𝑤𝐴𝑧𝐴 ((𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ∧ [𝑧 / 𝑤]𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑤 = 𝑧)))
208159, 206, 207sylanbrc 583 . . . . . . . . . . . . . 14 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → ∃!𝑤𝐴 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶))
209 riota1 7335 . . . . . . . . . . . . . 14 (∃!𝑤𝐴 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) → ((𝑤𝐴𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) ↔ (𝑤𝐴 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) = 𝑤))
210208, 209syl 17 . . . . . . . . . . . . 13 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → ((𝑤𝐴𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) ↔ (𝑤𝐴 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) = 𝑤))
211128, 157, 210mpbi2and 710 . . . . . . . . . . . 12 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → (𝑤𝐴 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) = 𝑤)
212156, 211eqtr2id 2789 . . . . . . . . . . 11 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑤 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶)))
213152, 212jca 512 . . . . . . . . . 10 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → (𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶) ∧ 𝑤 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶))))
214213ex 413 . . . . . . . . 9 ((𝜑𝑤𝐴) → (𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) → (𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶) ∧ 𝑤 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶)))))
215125, 214sylan2 593 . . . . . . . 8 ((𝜑𝑤 ∈ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅}) → (𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) → (𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶) ∧ 𝑤 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶)))))
216215eximdv 1920 . . . . . . 7 ((𝜑𝑤 ∈ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅}) → (∃𝑦 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) → ∃𝑦(𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶) ∧ 𝑤 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶)))))
217124, 216mpd 15 . . . . . 6 ((𝜑𝑤 ∈ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅}) → ∃𝑦(𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶) ∧ 𝑤 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶))))
218 df-rex 3074 . . . . . 6 (∃𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶)𝑤 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶)) ↔ ∃𝑦(𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶) ∧ 𝑤 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶))))
219217, 218sylibr 233 . . . . 5 ((𝜑𝑤 ∈ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅}) → ∃𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶)𝑤 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶)))
220219ralrimiva 3143 . . . 4 (𝜑 → ∀𝑤 ∈ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅}∃𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶)𝑤 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶)))
221 eqid 2736 . . . . 5 (𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶) ↦ (𝑥𝐴 𝑦 ∈ (𝐵𝐶))) = (𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶) ↦ (𝑥𝐴 𝑦 ∈ (𝐵𝐶)))
222221fompt 43401 . . . 4 ((𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶) ↦ (𝑥𝐴 𝑦 ∈ (𝐵𝐶))):( ran (𝑥𝐴𝐵) ∩ 𝐶)–onto→{𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅} ↔ (∀𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶)(𝑥𝐴 𝑦 ∈ (𝐵𝐶)) ∈ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅} ∧ ∀𝑤 ∈ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅}∃𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶)𝑤 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶))))
223114, 220, 222sylanbrc 583 . . 3 (𝜑 → (𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶) ↦ (𝑥𝐴 𝑦 ∈ (𝐵𝐶))):( ran (𝑥𝐴𝐵) ∩ 𝐶)–onto→{𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅})
224 fodomg 10458 . . 3 (( ran (𝑥𝐴𝐵) ∩ 𝐶) ∈ V → ((𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶) ↦ (𝑥𝐴 𝑦 ∈ (𝐵𝐶))):( ran (𝑥𝐴𝐵) ∩ 𝐶)–onto→{𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅} → {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅} ≼ ( ran (𝑥𝐴𝐵) ∩ 𝐶)))
2256, 223, 224sylc 65 . 2 (𝜑 → {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅} ≼ ( ran (𝑥𝐴𝐵) ∩ 𝐶))
226 domfi 9136 . 2 ((( ran (𝑥𝐴𝐵) ∩ 𝐶) ∈ Fin ∧ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅} ≼ ( ran (𝑥𝐴𝐵) ∩ 𝐶)) → {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅} ∈ Fin)
2274, 225, 226syl2anc 584 1 (𝜑 → {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅} ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wex 1781  [wsb 2067  wcel 2106  wne 2943  wral 3064  wrex 3073  ∃!wreu 3351  {crab 3407  Vcvv 3445  [wsbc 3739  csb 3855  cin 3909  wss 3910  c0 4282   cuni 4865  Disj wdisj 5070   class class class wbr 5105  cmpt 5188  ran crn 5634  ontowfo 6494  crio 7312  cdom 8881  Fincfn 8883
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-ac2 10399
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-fin 8887  df-card 9875  df-acn 9878  df-ac 10052
This theorem is referenced by:  fsumiunss  43806  sge0iunmptlemre  44646
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