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Theorem disjinfi 43970
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 4229 . . 3 ( ran (𝑥𝐴𝐵) ∩ 𝐶) ⊆ 𝐶
3 ssfi 9175 . . 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 5324 . . 3 (𝜑 → ( ran (𝑥𝐴𝐵) ∩ 𝐶) ∈ V)
7 elinel1 4195 . . . . . . . . . 10 (𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶) → 𝑦 ran (𝑥𝐴𝐵))
8 eluni2 4912 . . . . . . . . . . . 12 (𝑦 ran (𝑥𝐴𝐵) ↔ ∃𝑤 ∈ ran (𝑥𝐴𝐵)𝑦𝑤)
98biimpi 215 . . . . . . . . . . 11 (𝑦 ran (𝑥𝐴𝐵) → ∃𝑤 ∈ ran (𝑥𝐴𝐵)𝑦𝑤)
10 eqid 2732 . . . . . . . . . . . . . . . . . . 19 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
1110elrnmpt 5955 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ V → (𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑤 = 𝐵))
1211elv 3480 . . . . . . . . . . . . . . . . 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 5256 . . . . . . . . . . . . . . . . . . 19 𝑥(𝑥𝐴𝐵)
1615nfrn 5951 . . . . . . . . . . . . . . . . . 18 𝑥ran (𝑥𝐴𝐵)
1716nfcri 2890 . . . . . . . . . . . . . . . . 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 2835 . . . . . . . . . . . . . . . . . . 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 3258 . . . . . . . . . . . . . . 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 3153 . . . . . . . . . . 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 4915 . . . . . . . . . . . 12 𝑥 ran (𝑥𝐴𝐵)
36 nfcv 2903 . . . . . . . . . . . 12 𝑥𝐶
3735, 36nfin 4216 . . . . . . . . . . 11 𝑥( ran (𝑥𝐴𝐵) ∩ 𝐶)
3837nfcri 2890 . . . . . . . . . 10 𝑥 𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶)
3934, 38nfan 1902 . . . . . . . . 9 𝑥(𝜑𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶))
40 nfre1 3282 . . . . . . . . 9 𝑥𝑥𝐴 𝑦 ∈ (𝐵𝐶)
41 elinel2 4196 . . . . . . . . . . 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 4194 . . . . . . . . . . . . 13 ((𝑦𝐶𝑦𝐵) → 𝑦 ∈ (𝐵𝐶))
46 rspe 3246 . . . . . . . . . . . . 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 3263 . . . . . . . 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 5129 . . . . . . . . . . . . . . 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 2903 . . . . . . . . . . . . . . . 16 𝑥𝐴
58 nfv 1917 . . . . . . . . . . . . . . . . 17 𝑥 𝑧 = 𝑤
59 nfcsb1v 3918 . . . . . . . . . . . . . . . . . . 19 𝑥𝑧 / 𝑥𝐵
60 nfcv 2903 . . . . . . . . . . . . . . . . . . . 20 𝑥𝑤
6160nfcsb1 3917 . . . . . . . . . . . . . . . . . . 19 𝑥𝑤 / 𝑥𝐵
6259, 61nfin 4216 . . . . . . . . . . . . . . . . . 18 𝑥(𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵)
6362nfeq1 2918 . . . . . . . . . . . . . . . . 17 𝑥(𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵) = ∅
6458, 63nfor 1907 . . . . . . . . . . . . . . . 16 𝑥(𝑧 = 𝑤 ∨ (𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵) = ∅)
6557, 64nfralw 3308 . . . . . . . . . . . . . . 15 𝑥𝑤𝐴 (𝑧 = 𝑤 ∨ (𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵) = ∅)
66 equequ1 2028 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (𝑥 = 𝑤𝑧 = 𝑤))
67 csbeq1a 3907 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
6867ineq1d 4211 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑧 → (𝐵𝑤 / 𝑥𝐵) = (𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵))
6968eqeq1d 2734 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → ((𝐵𝑤 / 𝑥𝐵) = ∅ ↔ (𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵) = ∅))
7066, 69orbi12d 917 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → ((𝑥 = 𝑤 ∨ (𝐵𝑤 / 𝑥𝐵) = ∅) ↔ (𝑧 = 𝑤 ∨ (𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵) = ∅)))
7170ralbidv 3177 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (∀𝑤𝐴 (𝑥 = 𝑤 ∨ (𝐵𝑤 / 𝑥𝐵) = ∅) ↔ ∀𝑤𝐴 (𝑧 = 𝑤 ∨ (𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵) = ∅)))
7256, 65, 71cbvralw 3303 . . . . . . . . . . . . . 14 (∀𝑥𝐴𝑤𝐴 (𝑥 = 𝑤 ∨ (𝐵𝑤 / 𝑥𝐵) = ∅) ↔ ∀𝑧𝐴𝑤𝐴 (𝑧 = 𝑤 ∨ (𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵) = ∅))
7355, 72sylibr 233 . . . . . . . . . . . . 13 (𝜑 → ∀𝑥𝐴𝑤𝐴 (𝑥 = 𝑤 ∨ (𝐵𝑤 / 𝑥𝐵) = ∅))
7473r19.21bi 3248 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ∀𝑤𝐴 (𝑥 = 𝑤 ∨ (𝐵𝑤 / 𝑥𝐵) = ∅))
75 rspa 3245 . . . . . . . . . . . . 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 4195 . . . . . . . . . . . 12 (𝑦 ∈ (𝐵𝐶) → 𝑦𝐵)
79 sbsbc 3781 . . . . . . . . . . . . . 14 ([𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶) ↔ [𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶))
80 sbcel2 4415 . . . . . . . . . . . . . 14 ([𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶) ↔ 𝑦𝑤 / 𝑥(𝐵𝐶))
81 csbin 4439 . . . . . . . . . . . . . . 15 𝑤 / 𝑥(𝐵𝐶) = (𝑤 / 𝑥𝐵𝑤 / 𝑥𝐶)
8281eleq2i 2825 . . . . . . . . . . . . . 14 (𝑦𝑤 / 𝑥(𝐵𝐶) ↔ 𝑦 ∈ (𝑤 / 𝑥𝐵𝑤 / 𝑥𝐶))
8379, 80, 823bitri 296 . . . . . . . . . . . . 13 ([𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶) ↔ 𝑦 ∈ (𝑤 / 𝑥𝐵𝑤 / 𝑥𝐶))
84 elinel1 4195 . . . . . . . . . . . . 13 (𝑦 ∈ (𝑤 / 𝑥𝐵𝑤 / 𝑥𝐶) → 𝑦𝑤 / 𝑥𝐵)
8583, 84sylbi 216 . . . . . . . . . . . 12 ([𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶) → 𝑦𝑤 / 𝑥𝐵)
86 inelcm 4464 . . . . . . . . . . . . 13 ((𝑦𝐵𝑦𝑤 / 𝑥𝐵) → (𝐵𝑤 / 𝑥𝐵) ≠ ∅)
8786neneqd 2945 . . . . . . . . . . . 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 3146 . . . . . . . . 9 ((𝜑𝑥𝐴) → ∀𝑤𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ [𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶)) → 𝑥 = 𝑤))
9291ralrimiva 3146 . . . . . . . 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 3721 . . . . . . 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 7384 . . . . . 6 (∃!𝑥𝐴 𝑦 ∈ (𝐵𝐶) → (𝑥𝐴 𝑦 ∈ (𝐵𝐶)) ∈ {𝑥𝐴𝑦 ∈ (𝐵𝐶)})
97 nfriota1 7374 . . . . . . . . 9 𝑥(𝑥𝐴 𝑦 ∈ (𝐵𝐶))
9897nfcsb1 3917 . . . . . . . . . . 11 𝑥(𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵
9998, 36nfin 4216 . . . . . . . . . 10 𝑥((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵𝐶)
10099nfcri 2890 . . . . . . . . 9 𝑥 𝑦 ∈ ((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵𝐶)
101 csbeq1a 3907 . . . . . . . . . . 11 (𝑥 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶)) → 𝐵 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵)
102101ineq1d 4211 . . . . . . . . . 10 (𝑥 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶)) → (𝐵𝐶) = ((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵𝐶))
103102eleq2d 2819 . . . . . . . . 9 (𝑥 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶)) → (𝑦 ∈ (𝐵𝐶) ↔ 𝑦 ∈ ((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵𝐶)))
10497, 57, 100, 103elrabf 3679 . . . . . . . 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 4335 . . . . . . 7 ((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) ∈ {𝑥𝐴𝑦 ∈ (𝐵𝐶)} → ((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵𝐶) ≠ ∅)
108 nfcv 2903 . . . . . . . . 9 𝑥
10999, 108nfne 3043 . . . . . . . 8 𝑥((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵𝐶) ≠ ∅
110102neeq1d 3000 . . . . . . . 8 (𝑥 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶)) → ((𝐵𝐶) ≠ ∅ ↔ ((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵𝐶) ≠ ∅))
11197, 57, 109, 110elrabf 3679 . . . . . . 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 3146 . . . 4 (𝜑 → ∀𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶)(𝑥𝐴 𝑦 ∈ (𝐵𝐶)) ∈ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅})
11561, 36nfin 4216 . . . . . . . . . . . 12 𝑥(𝑤 / 𝑥𝐵𝐶)
116115, 108nfne 3043 . . . . . . . . . . 11 𝑥(𝑤 / 𝑥𝐵𝐶) ≠ ∅
117 csbeq1a 3907 . . . . . . . . . . . . 13 (𝑥 = 𝑤𝐵 = 𝑤 / 𝑥𝐵)
118117ineq1d 4211 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (𝐵𝐶) = (𝑤 / 𝑥𝐵𝐶))
119118neeq1d 3000 . . . . . . . . . . 11 (𝑥 = 𝑤 → ((𝐵𝐶) ≠ ∅ ↔ (𝑤 / 𝑥𝐵𝐶) ≠ ∅))
12060, 57, 116, 119elrabf 3679 . . . . . . . . . 10 (𝑤 ∈ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅} ↔ (𝑤𝐴 ∧ (𝑤 / 𝑥𝐵𝐶) ≠ ∅))
121120simprbi 497 . . . . . . . . 9 (𝑤 ∈ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅} → (𝑤 / 𝑥𝐵𝐶) ≠ ∅)
122 n0 4346 . . . . . . . . 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 4195 . . . . . . . . . . . . . 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 2919 . . . . . . . . . . . . . . . . . 18 𝑥𝑤 / 𝑥𝐵𝑉
131129, 130nfim 1899 . . . . . . . . . . . . . . . . 17 𝑥((𝜑𝑤𝐴) → 𝑤 / 𝑥𝐵𝑉)
132 eleq1w 2816 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑤 → (𝑥𝐴𝑤𝐴))
133132anbi2d 629 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑤 → ((𝜑𝑥𝐴) ↔ (𝜑𝑤𝐴)))
134117eleq1d 2818 . . . . . . . . . . . . . . . . . 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 2732 . . . . . . . . . . . . . . . 16 (𝑤𝐴𝑤 / 𝑥𝐵) = (𝑤𝐴𝑤 / 𝑥𝐵)
140139elrnmpt1 5957 . . . . . . . . . . . . . . 15 ((𝑤𝐴𝑤 / 𝑥𝐵𝑉) → 𝑤 / 𝑥𝐵 ∈ ran (𝑤𝐴𝑤 / 𝑥𝐵))
141128, 138, 140syl2anc 584 . . . . . . . . . . . . . 14 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑤 / 𝑥𝐵 ∈ ran (𝑤𝐴𝑤 / 𝑥𝐵))
142 nfcv 2903 . . . . . . . . . . . . . . . 16 𝑤𝐵
143117equcoms 2023 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑥𝐵 = 𝑤 / 𝑥𝐵)
144143eqcomd 2738 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑥𝑤 / 𝑥𝐵 = 𝐵)
14561, 142, 144cbvmpt 5259 . . . . . . . . . . . . . . 15 (𝑤𝐴𝑤 / 𝑥𝐵) = (𝑥𝐴𝐵)
146145rneqi 5936 . . . . . . . . . . . . . 14 ran (𝑤𝐴𝑤 / 𝑥𝐵) = ran (𝑥𝐴𝐵)
147141, 146eleqtrdi 2843 . . . . . . . . . . . . 13 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑤 / 𝑥𝐵 ∈ ran (𝑥𝐴𝐵))
148 elunii 4913 . . . . . . . . . . . . 13 ((𝑦𝑤 / 𝑥𝐵𝑤 / 𝑥𝐵 ∈ ran (𝑥𝐴𝐵)) → 𝑦 ran (𝑥𝐴𝐵))
149127, 147, 148syl2anc 584 . . . . . . . . . . . 12 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑦 ran (𝑥𝐴𝐵))
150 elinel2 4196 . . . . . . . . . . . . 13 (𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) → 𝑦𝐶)
151150adantl 482 . . . . . . . . . . . 12 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑦𝐶)
152149, 151elind 4194 . . . . . . . . . . 11 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶))
153 nfv 1917 . . . . . . . . . . . . 13 𝑤 𝑦 ∈ (𝐵𝐶)
154115nfcri 2890 . . . . . . . . . . . . 13 𝑥 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)
155118eleq2d 2819 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (𝑦 ∈ (𝐵𝐶) ↔ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)))
156153, 154, 155cbvriotaw 7376 . . . . . . . . . . . 12 (𝑥𝐴 𝑦 ∈ (𝐵𝐶)) = (𝑤𝐴 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶))
157 simpr 485 . . . . . . . . . . . . 13 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶))
158 rspe 3246 . . . . . . . . . . . . . . . 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 3781 . . . . . . . . . . . . . . . . . . . . . . . 24 ([𝑧 / 𝑥]𝑦 ∈ (𝐵𝐶) ↔ [𝑧 / 𝑥]𝑦 ∈ (𝐵𝐶))
163162a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑧 → ([𝑧 / 𝑥]𝑦 ∈ (𝐵𝐶) ↔ [𝑧 / 𝑥]𝑦 ∈ (𝐵𝐶)))
164 sbcel2 4415 . . . . . . . . . . . . . . . . . . . . . . . . 25 ([𝑧 / 𝑥]𝑦 ∈ (𝐵𝐶) ↔ 𝑦𝑧 / 𝑥(𝐵𝐶))
165 csbin 4439 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑧 / 𝑥(𝐵𝐶) = (𝑧 / 𝑥𝐵𝑧 / 𝑥𝐶)
166 csbconstg 3912 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 ∈ V → 𝑧 / 𝑥𝐶 = 𝐶)
167166elv 3480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑧 / 𝑥𝐶 = 𝐶
168167ineq2i 4209 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 / 𝑥𝐵𝑧 / 𝑥𝐶) = (𝑧 / 𝑥𝐵𝐶)
169165, 168eqtri 2760 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑧 / 𝑥(𝐵𝐶) = (𝑧 / 𝑥𝐵𝐶)
170169eleq2i 2825 . . . . . . . . . . . . . . . . . . . . . . . . 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 3234 . . . . . . . . . . . . . . . . . . 19 (∀𝑤𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ [𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶)) → 𝑥 = 𝑤) ↔ ∀𝑧𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑥 = 𝑧))
178177ralbii 3093 . . . . . . . . . . . . . . . . . 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 4216 . . . . . . . . . . . . . . . . . . . . . . 23 𝑥(𝑧 / 𝑥𝐵𝐶)
181180nfcri 2890 . . . . . . . . . . . . . . . . . . . . . 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 3308 . . . . . . . . . . . . . . . . . . 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 3177 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑤 → (∀𝑧𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑥 = 𝑧) ↔ ∀𝑧𝐴 ((𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑤 = 𝑧)))
190179, 185, 189cbvralw 3303 . . . . . . . . . . . . . . . . . 18 (∀𝑥𝐴𝑧𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑥 = 𝑧) ↔ ∀𝑤𝐴𝑧𝐴 ((𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑤 = 𝑧))
191 sbsbc 3781 . . . . . . . . . . . . . . . . . . . . . 22 ([𝑧 / 𝑤]𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ↔ [𝑧 / 𝑤]𝑦 ∈ (𝑤 / 𝑥𝐵𝐶))
192 sbcel2 4415 . . . . . . . . . . . . . . . . . . . . . 22 ([𝑧 / 𝑤]𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ↔ 𝑦𝑧 / 𝑤(𝑤 / 𝑥𝐵𝐶))
193 csbin 4439 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑧 / 𝑤(𝑤 / 𝑥𝐵𝐶) = (𝑧 / 𝑤𝑤 / 𝑥𝐵𝑧 / 𝑤𝐶)
194 csbcow 3908 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑧 / 𝑤𝑤 / 𝑥𝐵 = 𝑧 / 𝑥𝐵
195 csbconstg 3912 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 ∈ V → 𝑧 / 𝑤𝐶 = 𝐶)
196195elv 3480 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑧 / 𝑤𝐶 = 𝐶
197194, 196ineq12i 4210 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 / 𝑤𝑤 / 𝑥𝐵𝑧 / 𝑤𝐶) = (𝑧 / 𝑥𝐵𝐶)
198193, 197eqtri 2760 . . . . . . . . . . . . . . . . . . . . . . 23 𝑧 / 𝑤(𝑤 / 𝑥𝐵𝐶) = (𝑧 / 𝑥𝐵𝐶)
199198eleq2i 2825 . . . . . . . . . . . . . . . . . . . . . 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 3128 . . . . . . . . . . . . . . . . . 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 3721 . . . . . . . . . . . . . . 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 7389 . . . . . . . . . . . . . 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 2785 . . . . . . . . . . 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 3071 . . . . . 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 3146 . . . 4 (𝜑 → ∀𝑤 ∈ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅}∃𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶)𝑤 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶)))
221 eqid 2732 . . . . 5 (𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶) ↦ (𝑥𝐴 𝑦 ∈ (𝐵𝐶))) = (𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶) ↦ (𝑥𝐴 𝑦 ∈ (𝐵𝐶)))
222221fompt 7119 . . . 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 10519 . . 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 9194 . 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 2940  wral 3061  wrex 3070  ∃!wreu 3374  {crab 3432  Vcvv 3474  [wsbc 3777  csb 3893  cin 3947  wss 3948  c0 4322   cuni 4908  Disj wdisj 5113   class class class wbr 5148  cmpt 5231  ran crn 5677  ontowfo 6541  crio 7366  cdom 8939  Fincfn 8941
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-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-ac2 10460
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 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-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-disj 5114  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-1o 8468  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-fin 8945  df-card 9936  df-acn 9939  df-ac 10113
This theorem is referenced by:  fsumiunss  44370  sge0iunmptlemre  45210
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