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Theorem disjinfi 44190
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 9176 . . 3 ((𝐶 ∈ Fin ∧ ( ran (𝑥𝐴𝐵) ∩ 𝐶) ⊆ 𝐶) → ( ran (𝑥𝐴𝐵) ∩ 𝐶) ∈ Fin)
41, 2, 3sylancl 585 . 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 2731 . . . . . . . . . . . . . . . . . . 19 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
1110elrnmpt 5955 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ V → (𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑤 = 𝐵))
1211elv 3479 . . . . . . . . . . . . . . . . 17 (𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑤 = 𝐵)
1312biimpi 215 . . . . . . . . . . . . . . . 16 (𝑤 ∈ ran (𝑥𝐴𝐵) → ∃𝑥𝐴 𝑤 = 𝐵)
1413adantr 480 . . . . . . . . . . . . . . 15 ((𝑤 ∈ ran (𝑥𝐴𝐵) ∧ 𝑦𝑤) → ∃𝑥𝐴 𝑤 = 𝐵)
15 nfmpt1 5256 . . . . . . . . . . . . . . . . . . 19 𝑥(𝑥𝐴𝐵)
1615nfrn 5951 . . . . . . . . . . . . . . . . . 18 𝑥ran (𝑥𝐴𝐵)
1716nfcri 2889 . . . . . . . . . . . . . . . . 17 𝑥 𝑤 ∈ ran (𝑥𝐴𝐵)
18 nfv 1916 . . . . . . . . . . . . . . . . 17 𝑥 𝑦𝑤
1917, 18nfan 1901 . . . . . . . . . . . . . . . 16 𝑥(𝑤 ∈ ran (𝑥𝐴𝐵) ∧ 𝑦𝑤)
20 simpl 482 . . . . . . . . . . . . . . . . . . . 20 ((𝑦𝑤𝑤 = 𝐵) → 𝑦𝑤)
21 simpr 484 . . . . . . . . . . . . . . . . . . . 20 ((𝑦𝑤𝑤 = 𝐵) → 𝑤 = 𝐵)
2220, 21eleqtrd 2834 . . . . . . . . . . . . . . . . . . 19 ((𝑦𝑤𝑤 = 𝐵) → 𝑦𝐵)
2322ex 412 . . . . . . . . . . . . . . . . . 18 (𝑦𝑤 → (𝑤 = 𝐵𝑦𝐵))
2423a1d 25 . . . . . . . . . . . . . . . . 17 (𝑦𝑤 → (𝑥𝐴 → (𝑤 = 𝐵𝑦𝐵)))
2524adantl 481 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ ran (𝑥𝐴𝐵) ∧ 𝑦𝑤) → (𝑥𝐴 → (𝑤 = 𝐵𝑦𝐵)))
2619, 25reximdai 3257 . . . . . . . . . . . . . . 15 ((𝑤 ∈ ran (𝑥𝐴𝐵) ∧ 𝑦𝑤) → (∃𝑥𝐴 𝑤 = 𝐵 → ∃𝑥𝐴 𝑦𝐵))
2714, 26mpd 15 . . . . . . . . . . . . . 14 ((𝑤 ∈ ran (𝑥𝐴𝐵) ∧ 𝑦𝑤) → ∃𝑥𝐴 𝑦𝐵)
2827ex 412 . . . . . . . . . . . . 13 (𝑤 ∈ ran (𝑥𝐴𝐵) → (𝑦𝑤 → ∃𝑥𝐴 𝑦𝐵))
2928a1i 11 . . . . . . . . . . . 12 (𝑦 ran (𝑥𝐴𝐵) → (𝑤 ∈ ran (𝑥𝐴𝐵) → (𝑦𝑤 → ∃𝑥𝐴 𝑦𝐵)))
3029rexlimdv 3152 . . . . . . . . . . 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 481 . . . . . . . 8 ((𝜑𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶)) → ∃𝑥𝐴 𝑦𝐵)
34 nfv 1916 . . . . . . . . . 10 𝑥𝜑
3516nfuni 4915 . . . . . . . . . . . 12 𝑥 ran (𝑥𝐴𝐵)
36 nfcv 2902 . . . . . . . . . . . 12 𝑥𝐶
3735, 36nfin 4216 . . . . . . . . . . 11 𝑥( ran (𝑥𝐴𝐵) ∩ 𝐶)
3837nfcri 2889 . . . . . . . . . 10 𝑥 𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶)
3934, 38nfan 1901 . . . . . . . . 9 𝑥(𝜑𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶))
40 nfre1 3281 . . . . . . . . 9 𝑥𝑥𝐴 𝑦 ∈ (𝐵𝐶)
41 elinel2 4196 . . . . . . . . . . 11 (𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶) → 𝑦𝐶)
42 simp2 1136 . . . . . . . . . . . . 13 ((𝑦𝐶𝑥𝐴𝑦𝐵) → 𝑥𝐴)
43 simpr 484 . . . . . . . . . . . . . 14 ((𝑦𝐶𝑦𝐵) → 𝑦𝐵)
44 simpl 482 . . . . . . . . . . . . . 14 ((𝑦𝐶𝑦𝐵) → 𝑦𝐶)
4543, 44elind 4194 . . . . . . . . . . . . 13 ((𝑦𝐶𝑦𝐵) → 𝑦 ∈ (𝐵𝐶))
46 rspe 3245 . . . . . . . . . . . . 13 ((𝑥𝐴𝑦 ∈ (𝐵𝐶)) → ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
4742, 45, 463imp3i2an 1344 . . . . . . . . . . . 12 ((𝑦𝐶𝑥𝐴𝑦𝐵) → ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
48473exp 1118 . . . . . . . . . . 11 (𝑦𝐶 → (𝑥𝐴 → (𝑦𝐵 → ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))))
4941, 48syl 17 . . . . . . . . . 10 (𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶) → (𝑥𝐴 → (𝑦𝐵 → ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))))
5049adantl 481 . . . . . . . . 9 ((𝜑𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶)) → (𝑥𝐴 → (𝑦𝐵 → ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))))
5139, 40, 50rexlimd 3262 . . . . . . . 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 1916 . . . . . . . . . . . . . . 15 𝑧𝑤𝐴 (𝑥 = 𝑤 ∨ (𝐵𝑤 / 𝑥𝐵) = ∅)
57 nfcv 2902 . . . . . . . . . . . . . . . 16 𝑥𝐴
58 nfv 1916 . . . . . . . . . . . . . . . . 17 𝑥 𝑧 = 𝑤
59 nfcsb1v 3918 . . . . . . . . . . . . . . . . . . 19 𝑥𝑧 / 𝑥𝐵
60 nfcv 2902 . . . . . . . . . . . . . . . . . . . 20 𝑥𝑤
6160nfcsb1 3917 . . . . . . . . . . . . . . . . . . 19 𝑥𝑤 / 𝑥𝐵
6259, 61nfin 4216 . . . . . . . . . . . . . . . . . 18 𝑥(𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵)
6362nfeq1 2917 . . . . . . . . . . . . . . . . 17 𝑥(𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵) = ∅
6458, 63nfor 1906 . . . . . . . . . . . . . . . 16 𝑥(𝑧 = 𝑤 ∨ (𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵) = ∅)
6557, 64nfralw 3307 . . . . . . . . . . . . . . 15 𝑥𝑤𝐴 (𝑧 = 𝑤 ∨ (𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵) = ∅)
66 equequ1 2027 . . . . . . . . . . . . . . . . 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 2733 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → ((𝐵𝑤 / 𝑥𝐵) = ∅ ↔ (𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵) = ∅))
7066, 69orbi12d 916 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → ((𝑥 = 𝑤 ∨ (𝐵𝑤 / 𝑥𝐵) = ∅) ↔ (𝑧 = 𝑤 ∨ (𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵) = ∅)))
7170ralbidv 3176 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (∀𝑤𝐴 (𝑥 = 𝑤 ∨ (𝐵𝑤 / 𝑥𝐵) = ∅) ↔ ∀𝑤𝐴 (𝑧 = 𝑤 ∨ (𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵) = ∅)))
7256, 65, 71cbvralw 3302 . . . . . . . . . . . . . 14 (∀𝑥𝐴𝑤𝐴 (𝑥 = 𝑤 ∨ (𝐵𝑤 / 𝑥𝐵) = ∅) ↔ ∀𝑧𝐴𝑤𝐴 (𝑧 = 𝑤 ∨ (𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵) = ∅))
7355, 72sylibr 233 . . . . . . . . . . . . 13 (𝜑 → ∀𝑥𝐴𝑤𝐴 (𝑥 = 𝑤 ∨ (𝐵𝑤 / 𝑥𝐵) = ∅))
7473r19.21bi 3247 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ∀𝑤𝐴 (𝑥 = 𝑤 ∨ (𝐵𝑤 / 𝑥𝐵) = ∅))
75 rspa 3244 . . . . . . . . . . . . 13 ((∀𝑤𝐴 (𝑥 = 𝑤 ∨ (𝐵𝑤 / 𝑥𝐵) = ∅) ∧ 𝑤𝐴) → (𝑥 = 𝑤 ∨ (𝐵𝑤 / 𝑥𝐵) = ∅))
7675orcomd 868 . . . . . . . . . . . 12 ((∀𝑤𝐴 (𝑥 = 𝑤 ∨ (𝐵𝑤 / 𝑥𝐵) = ∅) ∧ 𝑤𝐴) → ((𝐵𝑤 / 𝑥𝐵) = ∅ ∨ 𝑥 = 𝑤))
7774, 76sylan 579 . . . . . . . . . . 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 2824 . . . . . . . . . . . . . 14 (𝑦𝑤 / 𝑥(𝐵𝐶) ↔ 𝑦 ∈ (𝑤 / 𝑥𝐵𝑤 / 𝑥𝐶))
8379, 80, 823bitri 297 . . . . . . . . . . . . 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 2944 . . . . . . . . . . . 12 ((𝑦𝐵𝑦𝑤 / 𝑥𝐵) → ¬ (𝐵𝑤 / 𝑥𝐵) = ∅)
8878, 85, 87syl2an 595 . . . . . . . . . . 11 ((𝑦 ∈ (𝐵𝐶) ∧ [𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶)) → ¬ (𝐵𝑤 / 𝑥𝐵) = ∅)
89 pm2.53 848 . . . . . . . . . . 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 3145 . . . . . . . . 9 ((𝜑𝑥𝐴) → ∀𝑤𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ [𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶)) → 𝑥 = 𝑤))
9291ralrimiva 3145 . . . . . . . 8 (𝜑 → ∀𝑥𝐴𝑤𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ [𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶)) → 𝑥 = 𝑤))
9392adantr 480 . . . . . . 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 582 . . . . . 6 ((𝜑𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶)) → ∃!𝑥𝐴 𝑦 ∈ (𝐵𝐶))
96 riotacl2 7385 . . . . . 6 (∃!𝑥𝐴 𝑦 ∈ (𝐵𝐶) → (𝑥𝐴 𝑦 ∈ (𝐵𝐶)) ∈ {𝑥𝐴𝑦 ∈ (𝐵𝐶)})
97 nfriota1 7375 . . . . . . . . 9 𝑥(𝑥𝐴 𝑦 ∈ (𝐵𝐶))
9897nfcsb1 3917 . . . . . . . . . . 11 𝑥(𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵
9998, 36nfin 4216 . . . . . . . . . 10 𝑥((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵𝐶)
10099nfcri 2889 . . . . . . . . 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 2818 . . . . . . . . 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 497 . . . . . . 7 ((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) ∈ {𝑥𝐴𝑦 ∈ (𝐵𝐶)} → (𝑥𝐴 𝑦 ∈ (𝐵𝐶)) ∈ 𝐴)
106104simprbi 496 . . . . . . . 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 2902 . . . . . . . . 9 𝑥
10999, 108nfne 3042 . . . . . . . 8 𝑥((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵𝐶) ≠ ∅
110102neeq1d 2999 . . . . . . . 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 582 . . . . . 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 3145 . . . 4 (𝜑 → ∀𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶)(𝑥𝐴 𝑦 ∈ (𝐵𝐶)) ∈ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅})
11561, 36nfin 4216 . . . . . . . . . . . 12 𝑥(𝑤 / 𝑥𝐵𝐶)
116115, 108nfne 3042 . . . . . . . . . . 11 𝑥(𝑤 / 𝑥𝐵𝐶) ≠ ∅
117 csbeq1a 3907 . . . . . . . . . . . . 13 (𝑥 = 𝑤𝐵 = 𝑤 / 𝑥𝐵)
118117ineq1d 4211 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (𝐵𝐶) = (𝑤 / 𝑥𝐵𝐶))
119118neeq1d 2999 . . . . . . . . . . 11 (𝑥 = 𝑤 → ((𝐵𝐶) ≠ ∅ ↔ (𝑤 / 𝑥𝐵𝐶) ≠ ∅))
12060, 57, 116, 119elrabf 3679 . . . . . . . . . 10 (𝑤 ∈ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅} ↔ (𝑤𝐴 ∧ (𝑤 / 𝑥𝐵𝐶) ≠ ∅))
121120simprbi 496 . . . . . . . . 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 481 . . . . . . 7 ((𝜑𝑤 ∈ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅}) → ∃𝑦 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶))
125120simplbi 497 . . . . . . . . 9 (𝑤 ∈ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅} → 𝑤𝐴)
126 elinel1 4195 . . . . . . . . . . . . . 14 (𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) → 𝑦𝑤 / 𝑥𝐵)
127126adantl 481 . . . . . . . . . . . . 13 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑦𝑤 / 𝑥𝐵)
128 simplr 766 . . . . . . . . . . . . . . 15 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑤𝐴)
129 nfv 1916 . . . . . . . . . . . . . . . . . 18 𝑥(𝜑𝑤𝐴)
13061nfel1 2918 . . . . . . . . . . . . . . . . . 18 𝑥𝑤 / 𝑥𝐵𝑉
131129, 130nfim 1898 . . . . . . . . . . . . . . . . 17 𝑥((𝜑𝑤𝐴) → 𝑤 / 𝑥𝐵𝑉)
132 eleq1w 2815 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑤 → (𝑥𝐴𝑤𝐴))
133132anbi2d 628 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑤 → ((𝜑𝑥𝐴) ↔ (𝜑𝑤𝐴)))
134117eleq1d 2817 . . . . . . . . . . . . . . . . . 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 2232 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝐴) → 𝑤 / 𝑥𝐵𝑉)
138137adantr 480 . . . . . . . . . . . . . . 15 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑤 / 𝑥𝐵𝑉)
139 eqid 2731 . . . . . . . . . . . . . . . 16 (𝑤𝐴𝑤 / 𝑥𝐵) = (𝑤𝐴𝑤 / 𝑥𝐵)
140139elrnmpt1 5957 . . . . . . . . . . . . . . 15 ((𝑤𝐴𝑤 / 𝑥𝐵𝑉) → 𝑤 / 𝑥𝐵 ∈ ran (𝑤𝐴𝑤 / 𝑥𝐵))
141128, 138, 140syl2anc 583 . . . . . . . . . . . . . 14 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑤 / 𝑥𝐵 ∈ ran (𝑤𝐴𝑤 / 𝑥𝐵))
142 nfcv 2902 . . . . . . . . . . . . . . . 16 𝑤𝐵
143117equcoms 2022 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑥𝐵 = 𝑤 / 𝑥𝐵)
144143eqcomd 2737 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑥𝑤 / 𝑥𝐵 = 𝐵)
14561, 142, 144cbvmpt 5259 . . . . . . . . . . . . . . 15 (𝑤𝐴𝑤 / 𝑥𝐵) = (𝑥𝐴𝐵)
146145rneqi 5936 . . . . . . . . . . . . . 14 ran (𝑤𝐴𝑤 / 𝑥𝐵) = ran (𝑥𝐴𝐵)
147141, 146eleqtrdi 2842 . . . . . . . . . . . . 13 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑤 / 𝑥𝐵 ∈ ran (𝑥𝐴𝐵))
148 elunii 4913 . . . . . . . . . . . . 13 ((𝑦𝑤 / 𝑥𝐵𝑤 / 𝑥𝐵 ∈ ran (𝑥𝐴𝐵)) → 𝑦 ran (𝑥𝐴𝐵))
149127, 147, 148syl2anc 583 . . . . . . . . . . . 12 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑦 ran (𝑥𝐴𝐵))
150 elinel2 4196 . . . . . . . . . . . . 13 (𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) → 𝑦𝐶)
151150adantl 481 . . . . . . . . . . . 12 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑦𝐶)
152149, 151elind 4194 . . . . . . . . . . 11 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶))
153 nfv 1916 . . . . . . . . . . . . 13 𝑤 𝑦 ∈ (𝐵𝐶)
154115nfcri 2889 . . . . . . . . . . . . 13 𝑥 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)
155118eleq2d 2818 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (𝑦 ∈ (𝐵𝐶) ↔ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)))
156153, 154, 155cbvriotaw 7377 . . . . . . . . . . . 12 (𝑥𝐴 𝑦 ∈ (𝐵𝐶)) = (𝑤𝐴 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶))
157 simpr 484 . . . . . . . . . . . . 13 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶))
158 rspe 3245 . . . . . . . . . . . . . . . 16 ((𝑤𝐴𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → ∃𝑤𝐴 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶))
159158adantll 711 . . . . . . . . . . . . . . 15 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → ∃𝑤𝐴 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶))
160 simpll 764 . . . . . . . . . . . . . . . 16 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝜑)
161 sbequ 2085 . . . . . . . . . . . . . . . . . . . . . . 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 3479 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑧 / 𝑥𝐶 = 𝐶
168167ineq2i 4209 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 / 𝑥𝐵𝑧 / 𝑥𝐶) = (𝑧 / 𝑥𝐵𝐶)
169165, 168eqtri 2759 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑧 / 𝑥(𝐵𝐶) = (𝑧 / 𝑥𝐵𝐶)
170169eleq2i 2824 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦𝑧 / 𝑥(𝐵𝐶) ↔ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶))
171164, 170bitri 275 . . . . . . . . . . . . . . . . . . . . . . . 24 ([𝑧 / 𝑥]𝑦 ∈ (𝐵𝐶) ↔ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶))
172171a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑧 → ([𝑧 / 𝑥]𝑦 ∈ (𝐵𝐶) ↔ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)))
173161, 163, 1723bitrd 305 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = 𝑧 → ([𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶) ↔ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)))
174173anbi2d 628 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑧 → ((𝑦 ∈ (𝐵𝐶) ∧ [𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶)) ↔ (𝑦 ∈ (𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶))))
175 equequ2 2028 . . . . . . . . . . . . . . . . . . . . 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 3233 . . . . . . . . . . . . . . . . . . 19 (∀𝑤𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ [𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶)) → 𝑥 = 𝑤) ↔ ∀𝑧𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑥 = 𝑧))
178177ralbii 3092 . . . . . . . . . . . . . . . . . 18 (∀𝑥𝐴𝑤𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ [𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶)) → 𝑥 = 𝑤) ↔ ∀𝑥𝐴𝑧𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑥 = 𝑧))
179 nfv 1916 . . . . . . . . . . . . . . . . . . 19 𝑤𝑧𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑥 = 𝑧)
18059, 36nfin 4216 . . . . . . . . . . . . . . . . . . . . . . 23 𝑥(𝑧 / 𝑥𝐵𝐶)
181180nfcri 2889 . . . . . . . . . . . . . . . . . . . . . 22 𝑥 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)
182154, 181nfan 1901 . . . . . . . . . . . . . . . . . . . . 21 𝑥(𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶))
183 nfv 1916 . . . . . . . . . . . . . . . . . . . . 21 𝑥 𝑤 = 𝑧
184182, 183nfim 1898 . . . . . . . . . . . . . . . . . . . 20 𝑥((𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑤 = 𝑧)
18557, 184nfralw 3307 . . . . . . . . . . . . . . . . . . 19 𝑥𝑧𝐴 ((𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑤 = 𝑧)
186155anbi1d 629 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑤 → ((𝑦 ∈ (𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) ↔ (𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶))))
187 equequ1 2027 . . . . . . . . . . . . . . . . . . . . 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 3176 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑤 → (∀𝑧𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑥 = 𝑧) ↔ ∀𝑧𝐴 ((𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑤 = 𝑧)))
190179, 185, 189cbvralw 3302 . . . . . . . . . . . . . . . . . 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 3479 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑧 / 𝑤𝐶 = 𝐶
197194, 196ineq12i 4210 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 / 𝑤𝑤 / 𝑥𝐵𝑧 / 𝑤𝐶) = (𝑧 / 𝑥𝐵𝐶)
198193, 197eqtri 2759 . . . . . . . . . . . . . . . . . . . . . . 23 𝑧 / 𝑤(𝑤 / 𝑥𝐵𝐶) = (𝑧 / 𝑥𝐵𝐶)
199198eleq2i 2824 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦𝑧 / 𝑤(𝑤 / 𝑥𝐵𝐶) ↔ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶))
200191, 192, 1993bitrri 298 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (𝑧 / 𝑥𝐵𝐶) ↔ [𝑧 / 𝑤]𝑦 ∈ (𝑤 / 𝑥𝐵𝐶))
201200anbi2i 622 . . . . . . . . . . . . . . . . . . . 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 297 . . . . . . . . . . . . . . . . 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 583 . . . . . . . . . . . . . . 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 582 . . . . . . . . . . . . . 14 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → ∃!𝑤𝐴 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶))
209 riota1 7390 . . . . . . . . . . . . . 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 709 . . . . . . . . . . . 12 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → (𝑤𝐴 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) = 𝑤)
212156, 211eqtr2id 2784 . . . . . . . . . . 11 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑤 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶)))
213152, 212jca 511 . . . . . . . . . 10 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → (𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶) ∧ 𝑤 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶))))
214213ex 412 . . . . . . . . 9 ((𝜑𝑤𝐴) → (𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) → (𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶) ∧ 𝑤 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶)))))
215125, 214sylan2 592 . . . . . . . 8 ((𝜑𝑤 ∈ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅}) → (𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) → (𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶) ∧ 𝑤 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶)))))
216215eximdv 1919 . . . . . . 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 3070 . . . . . 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 3145 . . . 4 (𝜑 → ∀𝑤 ∈ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅}∃𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶)𝑤 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶)))
221 eqid 2731 . . . . 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 582 . . 3 (𝜑 → (𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶) ↦ (𝑥𝐴 𝑦 ∈ (𝐵𝐶))):( ran (𝑥𝐴𝐵) ∩ 𝐶)–onto→{𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅})
224 fodomg 10520 . . 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 9195 . 2 ((( ran (𝑥𝐴𝐵) ∩ 𝐶) ∈ Fin ∧ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅} ≼ ( ran (𝑥𝐴𝐵) ∩ 𝐶)) → {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅} ∈ Fin)
2274, 225, 226syl2anc 583 1 (𝜑 → {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅} ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wo 844   = wceq 1540  wex 1780  [wsb 2066  wcel 2105  wne 2939  wral 3060  wrex 3069  ∃!wreu 3373  {crab 3431  Vcvv 3473  [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 7367  cdom 8940  Fincfn 8942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728  ax-ac2 10461
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  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 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-1o 8469  df-er 8706  df-map 8825  df-en 8943  df-dom 8944  df-fin 8946  df-card 9937  df-acn 9940  df-ac 10114
This theorem is referenced by:  fsumiunss  44590  sge0iunmptlemre  45430
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