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Theorem disjinfi 43348
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 4187 . . 3 ( ran (𝑥𝐴𝐵) ∩ 𝐶) ⊆ 𝐶
3 ssfi 9113 . . 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 5279 . . 3 (𝜑 → ( ran (𝑥𝐴𝐵) ∩ 𝐶) ∈ V)
7 elinel1 4153 . . . . . . . . . 10 (𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶) → 𝑦 ran (𝑥𝐴𝐵))
8 eluni2 4867 . . . . . . . . . . . 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 5909 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ V → (𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑤 = 𝐵))
1211elv 3449 . . . . . . . . . . . . . . . . 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 5211 . . . . . . . . . . . . . . . . . . 19 𝑥(𝑥𝐴𝐵)
1615nfrn 5905 . . . . . . . . . . . . . . . . . 18 𝑥ran (𝑥𝐴𝐵)
1716nfcri 2892 . . . . . . . . . . . . . . . . 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 3242 . . . . . . . . . . . . . . 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 3148 . . . . . . . . . . 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 4870 . . . . . . . . . . . 12 𝑥 ran (𝑥𝐴𝐵)
36 nfcv 2905 . . . . . . . . . . . 12 𝑥𝐶
3735, 36nfin 4174 . . . . . . . . . . 11 𝑥( ran (𝑥𝐴𝐵) ∩ 𝐶)
3837nfcri 2892 . . . . . . . . . 10 𝑥 𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶)
3934, 38nfan 1902 . . . . . . . . 9 𝑥(𝜑𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶))
40 nfre1 3266 . . . . . . . . 9 𝑥𝑥𝐴 𝑦 ∈ (𝐵𝐶)
41 elinel2 4154 . . . . . . . . . . 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 4152 . . . . . . . . . . . . 13 ((𝑦𝐶𝑦𝐵) → 𝑦 ∈ (𝐵𝐶))
46 rspe 3230 . . . . . . . . . . . . 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 3247 . . . . . . . 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 5084 . . . . . . . . . . . . . . 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 2905 . . . . . . . . . . . . . . . 16 𝑥𝐴
58 nfv 1917 . . . . . . . . . . . . . . . . 17 𝑥 𝑧 = 𝑤
59 nfcsb1v 3878 . . . . . . . . . . . . . . . . . . 19 𝑥𝑧 / 𝑥𝐵
60 nfcv 2905 . . . . . . . . . . . . . . . . . . . 20 𝑥𝑤
6160nfcsb1 3877 . . . . . . . . . . . . . . . . . . 19 𝑥𝑤 / 𝑥𝐵
6259, 61nfin 4174 . . . . . . . . . . . . . . . . . 18 𝑥(𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵)
6362nfeq1 2920 . . . . . . . . . . . . . . . . 17 𝑥(𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵) = ∅
6458, 63nfor 1907 . . . . . . . . . . . . . . . 16 𝑥(𝑧 = 𝑤 ∨ (𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵) = ∅)
6557, 64nfralw 3292 . . . . . . . . . . . . . . 15 𝑥𝑤𝐴 (𝑧 = 𝑤 ∨ (𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵) = ∅)
66 equequ1 2028 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (𝑥 = 𝑤𝑧 = 𝑤))
67 csbeq1a 3867 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
6867ineq1d 4169 . . . . . . . . . . . . . . . . . 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 3172 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (∀𝑤𝐴 (𝑥 = 𝑤 ∨ (𝐵𝑤 / 𝑥𝐵) = ∅) ↔ ∀𝑤𝐴 (𝑧 = 𝑤 ∨ (𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵) = ∅)))
7256, 65, 71cbvralw 3287 . . . . . . . . . . . . . 14 (∀𝑥𝐴𝑤𝐴 (𝑥 = 𝑤 ∨ (𝐵𝑤 / 𝑥𝐵) = ∅) ↔ ∀𝑧𝐴𝑤𝐴 (𝑧 = 𝑤 ∨ (𝑧 / 𝑥𝐵𝑤 / 𝑥𝐵) = ∅))
7355, 72sylibr 233 . . . . . . . . . . . . 13 (𝜑 → ∀𝑥𝐴𝑤𝐴 (𝑥 = 𝑤 ∨ (𝐵𝑤 / 𝑥𝐵) = ∅))
7473r19.21bi 3232 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ∀𝑤𝐴 (𝑥 = 𝑤 ∨ (𝐵𝑤 / 𝑥𝐵) = ∅))
75 rspa 3229 . . . . . . . . . . . . 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 4153 . . . . . . . . . . . 12 (𝑦 ∈ (𝐵𝐶) → 𝑦𝐵)
79 sbsbc 3741 . . . . . . . . . . . . . 14 ([𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶) ↔ [𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶))
80 sbcel2 4373 . . . . . . . . . . . . . 14 ([𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶) ↔ 𝑦𝑤 / 𝑥(𝐵𝐶))
81 csbin 4397 . . . . . . . . . . . . . . 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 4153 . . . . . . . . . . . . 13 (𝑦 ∈ (𝑤 / 𝑥𝐵𝑤 / 𝑥𝐶) → 𝑦𝑤 / 𝑥𝐵)
8583, 84sylbi 216 . . . . . . . . . . . 12 ([𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶) → 𝑦𝑤 / 𝑥𝐵)
86 inelcm 4422 . . . . . . . . . . . . 13 ((𝑦𝐵𝑦𝑤 / 𝑥𝐵) → (𝐵𝑤 / 𝑥𝐵) ≠ ∅)
8786neneqd 2946 . . . . . . . . . . . 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 3141 . . . . . . . . 9 ((𝜑𝑥𝐴) → ∀𝑤𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ [𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶)) → 𝑥 = 𝑤))
9291ralrimiva 3141 . . . . . . . 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 3681 . . . . . . 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 7326 . . . . . 6 (∃!𝑥𝐴 𝑦 ∈ (𝐵𝐶) → (𝑥𝐴 𝑦 ∈ (𝐵𝐶)) ∈ {𝑥𝐴𝑦 ∈ (𝐵𝐶)})
97 nfriota1 7316 . . . . . . . . 9 𝑥(𝑥𝐴 𝑦 ∈ (𝐵𝐶))
9897nfcsb1 3877 . . . . . . . . . . 11 𝑥(𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵
9998, 36nfin 4174 . . . . . . . . . 10 𝑥((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵𝐶)
10099nfcri 2892 . . . . . . . . 9 𝑥 𝑦 ∈ ((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵𝐶)
101 csbeq1a 3867 . . . . . . . . . . 11 (𝑥 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶)) → 𝐵 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵)
102101ineq1d 4169 . . . . . . . . . 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 3639 . . . . . . . 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 4293 . . . . . . 7 ((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) ∈ {𝑥𝐴𝑦 ∈ (𝐵𝐶)} → ((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵𝐶) ≠ ∅)
108 nfcv 2905 . . . . . . . . 9 𝑥
10999, 108nfne 3043 . . . . . . . 8 𝑥((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵𝐶) ≠ ∅
110102neeq1d 3001 . . . . . . . 8 (𝑥 = (𝑥𝐴 𝑦 ∈ (𝐵𝐶)) → ((𝐵𝐶) ≠ ∅ ↔ ((𝑥𝐴 𝑦 ∈ (𝐵𝐶)) / 𝑥𝐵𝐶) ≠ ∅))
11197, 57, 109, 110elrabf 3639 . . . . . . 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 3141 . . . 4 (𝜑 → ∀𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶)(𝑥𝐴 𝑦 ∈ (𝐵𝐶)) ∈ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅})
11561, 36nfin 4174 . . . . . . . . . . . 12 𝑥(𝑤 / 𝑥𝐵𝐶)
116115, 108nfne 3043 . . . . . . . . . . 11 𝑥(𝑤 / 𝑥𝐵𝐶) ≠ ∅
117 csbeq1a 3867 . . . . . . . . . . . . 13 (𝑥 = 𝑤𝐵 = 𝑤 / 𝑥𝐵)
118117ineq1d 4169 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (𝐵𝐶) = (𝑤 / 𝑥𝐵𝐶))
119118neeq1d 3001 . . . . . . . . . . 11 (𝑥 = 𝑤 → ((𝐵𝐶) ≠ ∅ ↔ (𝑤 / 𝑥𝐵𝐶) ≠ ∅))
12060, 57, 116, 119elrabf 3639 . . . . . . . . . 10 (𝑤 ∈ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅} ↔ (𝑤𝐴 ∧ (𝑤 / 𝑥𝐵𝐶) ≠ ∅))
121120simprbi 497 . . . . . . . . 9 (𝑤 ∈ {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅} → (𝑤 / 𝑥𝐵𝐶) ≠ ∅)
122 n0 4304 . . . . . . . . 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 4153 . . . . . . . . . . . . . 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 2921 . . . . . . . . . . . . . . . . . 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 5911 . . . . . . . . . . . . . . 15 ((𝑤𝐴𝑤 / 𝑥𝐵𝑉) → 𝑤 / 𝑥𝐵 ∈ ran (𝑤𝐴𝑤 / 𝑥𝐵))
141128, 138, 140syl2anc 584 . . . . . . . . . . . . . 14 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑤 / 𝑥𝐵 ∈ ran (𝑤𝐴𝑤 / 𝑥𝐵))
142 nfcv 2905 . . . . . . . . . . . . . . . 16 𝑤𝐵
143117equcoms 2023 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑥𝐵 = 𝑤 / 𝑥𝐵)
144143eqcomd 2742 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑥𝑤 / 𝑥𝐵 = 𝐵)
14561, 142, 144cbvmpt 5214 . . . . . . . . . . . . . . 15 (𝑤𝐴𝑤 / 𝑥𝐵) = (𝑥𝐴𝐵)
146145rneqi 5890 . . . . . . . . . . . . . 14 ran (𝑤𝐴𝑤 / 𝑥𝐵) = ran (𝑥𝐴𝐵)
147141, 146eleqtrdi 2848 . . . . . . . . . . . . 13 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑤 / 𝑥𝐵 ∈ ran (𝑥𝐴𝐵))
148 elunii 4868 . . . . . . . . . . . . 13 ((𝑦𝑤 / 𝑥𝐵𝑤 / 𝑥𝐵 ∈ ran (𝑥𝐴𝐵)) → 𝑦 ran (𝑥𝐴𝐵))
149127, 147, 148syl2anc 584 . . . . . . . . . . . 12 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑦 ran (𝑥𝐴𝐵))
150 elinel2 4154 . . . . . . . . . . . . 13 (𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) → 𝑦𝐶)
151150adantl 482 . . . . . . . . . . . 12 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑦𝐶)
152149, 151elind 4152 . . . . . . . . . . 11 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑦 ∈ ( ran (𝑥𝐴𝐵) ∩ 𝐶))
153 nfv 1917 . . . . . . . . . . . . 13 𝑤 𝑦 ∈ (𝐵𝐶)
154115nfcri 2892 . . . . . . . . . . . . 13 𝑥 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)
155118eleq2d 2823 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (𝑦 ∈ (𝐵𝐶) ↔ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)))
156153, 154, 155cbvriotaw 7318 . . . . . . . . . . . 12 (𝑥𝐴 𝑦 ∈ (𝐵𝐶)) = (𝑤𝐴 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶))
157 simpr 485 . . . . . . . . . . . . 13 (((𝜑𝑤𝐴) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶)) → 𝑦 ∈ (𝑤 / 𝑥𝐵𝐶))
158 rspe 3230 . . . . . . . . . . . . . . . 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 3741 . . . . . . . . . . . . . . . . . . . . . . . 24 ([𝑧 / 𝑥]𝑦 ∈ (𝐵𝐶) ↔ [𝑧 / 𝑥]𝑦 ∈ (𝐵𝐶))
163162a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑧 → ([𝑧 / 𝑥]𝑦 ∈ (𝐵𝐶) ↔ [𝑧 / 𝑥]𝑦 ∈ (𝐵𝐶)))
164 sbcel2 4373 . . . . . . . . . . . . . . . . . . . . . . . . 25 ([𝑧 / 𝑥]𝑦 ∈ (𝐵𝐶) ↔ 𝑦𝑧 / 𝑥(𝐵𝐶))
165 csbin 4397 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑧 / 𝑥(𝐵𝐶) = (𝑧 / 𝑥𝐵𝑧 / 𝑥𝐶)
166 csbconstg 3872 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 ∈ V → 𝑧 / 𝑥𝐶 = 𝐶)
167166elv 3449 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑧 / 𝑥𝐶 = 𝐶
168167ineq2i 4167 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3223 . . . . . . . . . . . . . . . . . . 19 (∀𝑤𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ [𝑤 / 𝑥]𝑦 ∈ (𝐵𝐶)) → 𝑥 = 𝑤) ↔ ∀𝑧𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑥 = 𝑧))
178177ralbii 3094 . . . . . . . . . . . . . . . . . 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 4174 . . . . . . . . . . . . . . . . . . . . . . 23 𝑥(𝑧 / 𝑥𝐵𝐶)
181180nfcri 2892 . . . . . . . . . . . . . . . . . . . . . 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 3292 . . . . . . . . . . . . . . . . . . 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 3172 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑤 → (∀𝑧𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑥 = 𝑧) ↔ ∀𝑧𝐴 ((𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑤 = 𝑧)))
190179, 185, 189cbvralw 3287 . . . . . . . . . . . . . . . . . 18 (∀𝑥𝐴𝑧𝐴 ((𝑦 ∈ (𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑥 = 𝑧) ↔ ∀𝑤𝐴𝑧𝐴 ((𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ∧ 𝑦 ∈ (𝑧 / 𝑥𝐵𝐶)) → 𝑤 = 𝑧))
191 sbsbc 3741 . . . . . . . . . . . . . . . . . . . . . 22 ([𝑧 / 𝑤]𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ↔ [𝑧 / 𝑤]𝑦 ∈ (𝑤 / 𝑥𝐵𝐶))
192 sbcel2 4373 . . . . . . . . . . . . . . . . . . . . . 22 ([𝑧 / 𝑤]𝑦 ∈ (𝑤 / 𝑥𝐵𝐶) ↔ 𝑦𝑧 / 𝑤(𝑤 / 𝑥𝐵𝐶))
193 csbin 4397 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑧 / 𝑤(𝑤 / 𝑥𝐵𝐶) = (𝑧 / 𝑤𝑤 / 𝑥𝐵𝑧 / 𝑤𝐶)
194 csbcow 3868 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑧 / 𝑤𝑤 / 𝑥𝐵 = 𝑧 / 𝑥𝐵
195 csbconstg 3872 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 ∈ V → 𝑧 / 𝑤𝐶 = 𝐶)
196195elv 3449 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑧 / 𝑤𝐶 = 𝐶
197194, 196ineq12i 4168 . . . . . . . . . . . . . . . . . . . . . . . 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 3125 . . . . . . . . . . . . . . . . . 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 3681 . . . . . . . . . . . . . . 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 7331 . . . . . . . . . . . . . 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 3072 . . . . . 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 3141 . . . 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 43347 . . . 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 10454 . . 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 9132 . 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 2941  wral 3062  wrex 3071  ∃!wreu 3349  {crab 3405  Vcvv 3443  [wsbc 3737  csb 3853  cin 3907  wss 3908  c0 4280   cuni 4863  Disj wdisj 5068   class class class wbr 5103  cmpt 5186  ran crn 5632  ontowfo 6491  crio 7308  cdom 8877  Fincfn 8879
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 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668  ax-ac2 10395
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 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-int 4906  df-iun 4954  df-disj 5069  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-se 5587  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7309  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7799  df-1st 7917  df-2nd 7918  df-frecs 8208  df-wrecs 8239  df-recs 8313  df-1o 8408  df-er 8644  df-map 8763  df-en 8880  df-dom 8881  df-fin 8883  df-card 9871  df-acn 9874  df-ac 10048
This theorem is referenced by:  fsumiunss  43748  sge0iunmptlemre  44588
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