Metamath Proof Explorer


Theorem disjinfi

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)

Ref Expression
Hypotheses disjinfi.b ( ( 𝜑𝑥𝐴 ) → 𝐵𝑉 )
disjinfi.d ( 𝜑Disj 𝑥𝐴 𝐵 )
disjinfi.c ( 𝜑𝐶 ∈ Fin )
Assertion disjinfi ( 𝜑 → { 𝑥𝐴 ∣ ( 𝐵𝐶 ) ≠ ∅ } ∈ Fin )

Proof

Step Hyp Ref Expression
1 disjinfi.b ( ( 𝜑𝑥𝐴 ) → 𝐵𝑉 )
2 disjinfi.d ( 𝜑Disj 𝑥𝐴 𝐵 )
3 disjinfi.c ( 𝜑𝐶 ∈ Fin )
4 inss2 ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) ⊆ 𝐶
5 ssfi ( ( 𝐶 ∈ Fin ∧ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) ⊆ 𝐶 ) → ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) ∈ Fin )
6 3 4 5 sylancl ( 𝜑 → ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) ∈ Fin )
7 4 a1i ( 𝜑 → ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) ⊆ 𝐶 )
8 3 7 ssexd ( 𝜑 → ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) ∈ V )
9 elinel1 ( 𝑦 ∈ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) → 𝑦 ran ( 𝑥𝐴𝐵 ) )
10 eluni2 ( 𝑦 ran ( 𝑥𝐴𝐵 ) ↔ ∃ 𝑤 ∈ ran ( 𝑥𝐴𝐵 ) 𝑦𝑤 )
11 10 biimpi ( 𝑦 ran ( 𝑥𝐴𝐵 ) → ∃ 𝑤 ∈ ran ( 𝑥𝐴𝐵 ) 𝑦𝑤 )
12 eqid ( 𝑥𝐴𝐵 ) = ( 𝑥𝐴𝐵 )
13 12 elrnmpt ( 𝑤 ∈ V → ( 𝑤 ∈ ran ( 𝑥𝐴𝐵 ) ↔ ∃ 𝑥𝐴 𝑤 = 𝐵 ) )
14 13 elv ( 𝑤 ∈ ran ( 𝑥𝐴𝐵 ) ↔ ∃ 𝑥𝐴 𝑤 = 𝐵 )
15 14 biimpi ( 𝑤 ∈ ran ( 𝑥𝐴𝐵 ) → ∃ 𝑥𝐴 𝑤 = 𝐵 )
16 15 adantr ( ( 𝑤 ∈ ran ( 𝑥𝐴𝐵 ) ∧ 𝑦𝑤 ) → ∃ 𝑥𝐴 𝑤 = 𝐵 )
17 nfmpt1 𝑥 ( 𝑥𝐴𝐵 )
18 17 nfrn 𝑥 ran ( 𝑥𝐴𝐵 )
19 18 nfcri 𝑥 𝑤 ∈ ran ( 𝑥𝐴𝐵 )
20 nfv 𝑥 𝑦𝑤
21 19 20 nfan 𝑥 ( 𝑤 ∈ ran ( 𝑥𝐴𝐵 ) ∧ 𝑦𝑤 )
22 simpl ( ( 𝑦𝑤𝑤 = 𝐵 ) → 𝑦𝑤 )
23 simpr ( ( 𝑦𝑤𝑤 = 𝐵 ) → 𝑤 = 𝐵 )
24 22 23 eleqtrd ( ( 𝑦𝑤𝑤 = 𝐵 ) → 𝑦𝐵 )
25 24 ex ( 𝑦𝑤 → ( 𝑤 = 𝐵𝑦𝐵 ) )
26 25 a1d ( 𝑦𝑤 → ( 𝑥𝐴 → ( 𝑤 = 𝐵𝑦𝐵 ) ) )
27 26 adantl ( ( 𝑤 ∈ ran ( 𝑥𝐴𝐵 ) ∧ 𝑦𝑤 ) → ( 𝑥𝐴 → ( 𝑤 = 𝐵𝑦𝐵 ) ) )
28 21 27 reximdai ( ( 𝑤 ∈ ran ( 𝑥𝐴𝐵 ) ∧ 𝑦𝑤 ) → ( ∃ 𝑥𝐴 𝑤 = 𝐵 → ∃ 𝑥𝐴 𝑦𝐵 ) )
29 16 28 mpd ( ( 𝑤 ∈ ran ( 𝑥𝐴𝐵 ) ∧ 𝑦𝑤 ) → ∃ 𝑥𝐴 𝑦𝐵 )
30 29 ex ( 𝑤 ∈ ran ( 𝑥𝐴𝐵 ) → ( 𝑦𝑤 → ∃ 𝑥𝐴 𝑦𝐵 ) )
31 30 a1i ( 𝑦 ran ( 𝑥𝐴𝐵 ) → ( 𝑤 ∈ ran ( 𝑥𝐴𝐵 ) → ( 𝑦𝑤 → ∃ 𝑥𝐴 𝑦𝐵 ) ) )
32 31 rexlimdv ( 𝑦 ran ( 𝑥𝐴𝐵 ) → ( ∃ 𝑤 ∈ ran ( 𝑥𝐴𝐵 ) 𝑦𝑤 → ∃ 𝑥𝐴 𝑦𝐵 ) )
33 11 32 mpd ( 𝑦 ran ( 𝑥𝐴𝐵 ) → ∃ 𝑥𝐴 𝑦𝐵 )
34 9 33 syl ( 𝑦 ∈ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) → ∃ 𝑥𝐴 𝑦𝐵 )
35 34 adantl ( ( 𝜑𝑦 ∈ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) ) → ∃ 𝑥𝐴 𝑦𝐵 )
36 nfv 𝑥 𝜑
37 18 nfuni 𝑥 ran ( 𝑥𝐴𝐵 )
38 nfcv 𝑥 𝐶
39 37 38 nfin 𝑥 ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 )
40 39 nfcri 𝑥 𝑦 ∈ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 )
41 36 40 nfan 𝑥 ( 𝜑𝑦 ∈ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) )
42 nfre1 𝑥𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 )
43 elinel2 ( 𝑦 ∈ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) → 𝑦𝐶 )
44 simp2 ( ( 𝑦𝐶𝑥𝐴𝑦𝐵 ) → 𝑥𝐴 )
45 simpr ( ( 𝑦𝐶𝑦𝐵 ) → 𝑦𝐵 )
46 simpl ( ( 𝑦𝐶𝑦𝐵 ) → 𝑦𝐶 )
47 45 46 elind ( ( 𝑦𝐶𝑦𝐵 ) → 𝑦 ∈ ( 𝐵𝐶 ) )
48 rspe ( ( 𝑥𝐴𝑦 ∈ ( 𝐵𝐶 ) ) → ∃ 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) )
49 44 47 48 3imp3i2an ( ( 𝑦𝐶𝑥𝐴𝑦𝐵 ) → ∃ 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) )
50 49 3exp ( 𝑦𝐶 → ( 𝑥𝐴 → ( 𝑦𝐵 → ∃ 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) ) )
51 43 50 syl ( 𝑦 ∈ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) → ( 𝑥𝐴 → ( 𝑦𝐵 → ∃ 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) ) )
52 51 adantl ( ( 𝜑𝑦 ∈ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) ) → ( 𝑥𝐴 → ( 𝑦𝐵 → ∃ 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) ) )
53 41 42 52 rexlimd ( ( 𝜑𝑦 ∈ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) ) → ( ∃ 𝑥𝐴 𝑦𝐵 → ∃ 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) )
54 35 53 mpd ( ( 𝜑𝑦 ∈ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) ) → ∃ 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) )
55 disjors ( Disj 𝑥𝐴 𝐵 ↔ ∀ 𝑧𝐴𝑤𝐴 ( 𝑧 = 𝑤 ∨ ( 𝑧 / 𝑥 𝐵 𝑤 / 𝑥 𝐵 ) = ∅ ) )
56 2 55 sylib ( 𝜑 → ∀ 𝑧𝐴𝑤𝐴 ( 𝑧 = 𝑤 ∨ ( 𝑧 / 𝑥 𝐵 𝑤 / 𝑥 𝐵 ) = ∅ ) )
57 nfv 𝑧𝑤𝐴 ( 𝑥 = 𝑤 ∨ ( 𝐵 𝑤 / 𝑥 𝐵 ) = ∅ )
58 nfcv 𝑥 𝐴
59 nfv 𝑥 𝑧 = 𝑤
60 nfcsb1v 𝑥 𝑧 / 𝑥 𝐵
61 nfcv 𝑥 𝑤
62 61 nfcsb1 𝑥 𝑤 / 𝑥 𝐵
63 60 62 nfin 𝑥 ( 𝑧 / 𝑥 𝐵 𝑤 / 𝑥 𝐵 )
64 63 nfeq1 𝑥 ( 𝑧 / 𝑥 𝐵 𝑤 / 𝑥 𝐵 ) = ∅
65 59 64 nfor 𝑥 ( 𝑧 = 𝑤 ∨ ( 𝑧 / 𝑥 𝐵 𝑤 / 𝑥 𝐵 ) = ∅ )
66 58 65 nfralw 𝑥𝑤𝐴 ( 𝑧 = 𝑤 ∨ ( 𝑧 / 𝑥 𝐵 𝑤 / 𝑥 𝐵 ) = ∅ )
67 equequ1 ( 𝑥 = 𝑧 → ( 𝑥 = 𝑤𝑧 = 𝑤 ) )
68 csbeq1a ( 𝑥 = 𝑧𝐵 = 𝑧 / 𝑥 𝐵 )
69 68 ineq1d ( 𝑥 = 𝑧 → ( 𝐵 𝑤 / 𝑥 𝐵 ) = ( 𝑧 / 𝑥 𝐵 𝑤 / 𝑥 𝐵 ) )
70 69 eqeq1d ( 𝑥 = 𝑧 → ( ( 𝐵 𝑤 / 𝑥 𝐵 ) = ∅ ↔ ( 𝑧 / 𝑥 𝐵 𝑤 / 𝑥 𝐵 ) = ∅ ) )
71 67 70 orbi12d ( 𝑥 = 𝑧 → ( ( 𝑥 = 𝑤 ∨ ( 𝐵 𝑤 / 𝑥 𝐵 ) = ∅ ) ↔ ( 𝑧 = 𝑤 ∨ ( 𝑧 / 𝑥 𝐵 𝑤 / 𝑥 𝐵 ) = ∅ ) ) )
72 71 ralbidv ( 𝑥 = 𝑧 → ( ∀ 𝑤𝐴 ( 𝑥 = 𝑤 ∨ ( 𝐵 𝑤 / 𝑥 𝐵 ) = ∅ ) ↔ ∀ 𝑤𝐴 ( 𝑧 = 𝑤 ∨ ( 𝑧 / 𝑥 𝐵 𝑤 / 𝑥 𝐵 ) = ∅ ) ) )
73 57 66 72 cbvralw ( ∀ 𝑥𝐴𝑤𝐴 ( 𝑥 = 𝑤 ∨ ( 𝐵 𝑤 / 𝑥 𝐵 ) = ∅ ) ↔ ∀ 𝑧𝐴𝑤𝐴 ( 𝑧 = 𝑤 ∨ ( 𝑧 / 𝑥 𝐵 𝑤 / 𝑥 𝐵 ) = ∅ ) )
74 56 73 sylibr ( 𝜑 → ∀ 𝑥𝐴𝑤𝐴 ( 𝑥 = 𝑤 ∨ ( 𝐵 𝑤 / 𝑥 𝐵 ) = ∅ ) )
75 74 r19.21bi ( ( 𝜑𝑥𝐴 ) → ∀ 𝑤𝐴 ( 𝑥 = 𝑤 ∨ ( 𝐵 𝑤 / 𝑥 𝐵 ) = ∅ ) )
76 rspa ( ( ∀ 𝑤𝐴 ( 𝑥 = 𝑤 ∨ ( 𝐵 𝑤 / 𝑥 𝐵 ) = ∅ ) ∧ 𝑤𝐴 ) → ( 𝑥 = 𝑤 ∨ ( 𝐵 𝑤 / 𝑥 𝐵 ) = ∅ ) )
77 76 orcomd ( ( ∀ 𝑤𝐴 ( 𝑥 = 𝑤 ∨ ( 𝐵 𝑤 / 𝑥 𝐵 ) = ∅ ) ∧ 𝑤𝐴 ) → ( ( 𝐵 𝑤 / 𝑥 𝐵 ) = ∅ ∨ 𝑥 = 𝑤 ) )
78 75 77 sylan ( ( ( 𝜑𝑥𝐴 ) ∧ 𝑤𝐴 ) → ( ( 𝐵 𝑤 / 𝑥 𝐵 ) = ∅ ∨ 𝑥 = 𝑤 ) )
79 elinel1 ( 𝑦 ∈ ( 𝐵𝐶 ) → 𝑦𝐵 )
80 sbsbc ( [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵𝐶 ) ↔ [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵𝐶 ) )
81 sbcel2 ( [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵𝐶 ) ↔ 𝑦 𝑤 / 𝑥 ( 𝐵𝐶 ) )
82 csbin 𝑤 / 𝑥 ( 𝐵𝐶 ) = ( 𝑤 / 𝑥 𝐵 𝑤 / 𝑥 𝐶 )
83 82 eleq2i ( 𝑦 𝑤 / 𝑥 ( 𝐵𝐶 ) ↔ 𝑦 ∈ ( 𝑤 / 𝑥 𝐵 𝑤 / 𝑥 𝐶 ) )
84 80 81 83 3bitri ( [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵𝐶 ) ↔ 𝑦 ∈ ( 𝑤 / 𝑥 𝐵 𝑤 / 𝑥 𝐶 ) )
85 elinel1 ( 𝑦 ∈ ( 𝑤 / 𝑥 𝐵 𝑤 / 𝑥 𝐶 ) → 𝑦 𝑤 / 𝑥 𝐵 )
86 84 85 sylbi ( [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵𝐶 ) → 𝑦 𝑤 / 𝑥 𝐵 )
87 inelcm ( ( 𝑦𝐵𝑦 𝑤 / 𝑥 𝐵 ) → ( 𝐵 𝑤 / 𝑥 𝐵 ) ≠ ∅ )
88 87 neneqd ( ( 𝑦𝐵𝑦 𝑤 / 𝑥 𝐵 ) → ¬ ( 𝐵 𝑤 / 𝑥 𝐵 ) = ∅ )
89 79 86 88 syl2an ( ( 𝑦 ∈ ( 𝐵𝐶 ) ∧ [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵𝐶 ) ) → ¬ ( 𝐵 𝑤 / 𝑥 𝐵 ) = ∅ )
90 pm2.53 ( ( ( 𝐵 𝑤 / 𝑥 𝐵 ) = ∅ ∨ 𝑥 = 𝑤 ) → ( ¬ ( 𝐵 𝑤 / 𝑥 𝐵 ) = ∅ → 𝑥 = 𝑤 ) )
91 78 89 90 syl2im ( ( ( 𝜑𝑥𝐴 ) ∧ 𝑤𝐴 ) → ( ( 𝑦 ∈ ( 𝐵𝐶 ) ∧ [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵𝐶 ) ) → 𝑥 = 𝑤 ) )
92 91 ralrimiva ( ( 𝜑𝑥𝐴 ) → ∀ 𝑤𝐴 ( ( 𝑦 ∈ ( 𝐵𝐶 ) ∧ [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵𝐶 ) ) → 𝑥 = 𝑤 ) )
93 92 ralrimiva ( 𝜑 → ∀ 𝑥𝐴𝑤𝐴 ( ( 𝑦 ∈ ( 𝐵𝐶 ) ∧ [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵𝐶 ) ) → 𝑥 = 𝑤 ) )
94 93 adantr ( ( 𝜑𝑦 ∈ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) ) → ∀ 𝑥𝐴𝑤𝐴 ( ( 𝑦 ∈ ( 𝐵𝐶 ) ∧ [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵𝐶 ) ) → 𝑥 = 𝑤 ) )
95 reu2 ( ∃! 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ↔ ( ∃ 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ∧ ∀ 𝑥𝐴𝑤𝐴 ( ( 𝑦 ∈ ( 𝐵𝐶 ) ∧ [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵𝐶 ) ) → 𝑥 = 𝑤 ) ) )
96 54 94 95 sylanbrc ( ( 𝜑𝑦 ∈ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) ) → ∃! 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) )
97 riotacl2 ( ∃! 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) → ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) ∈ { 𝑥𝐴𝑦 ∈ ( 𝐵𝐶 ) } )
98 nfriota1 𝑥 ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) )
99 98 nfcsb1 𝑥 ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) / 𝑥 𝐵
100 99 38 nfin 𝑥 ( ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) / 𝑥 𝐵𝐶 )
101 100 nfcri 𝑥 𝑦 ∈ ( ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) / 𝑥 𝐵𝐶 )
102 csbeq1a ( 𝑥 = ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) → 𝐵 = ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) / 𝑥 𝐵 )
103 102 ineq1d ( 𝑥 = ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) → ( 𝐵𝐶 ) = ( ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) / 𝑥 𝐵𝐶 ) )
104 103 eleq2d ( 𝑥 = ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) → ( 𝑦 ∈ ( 𝐵𝐶 ) ↔ 𝑦 ∈ ( ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) / 𝑥 𝐵𝐶 ) ) )
105 98 58 101 104 elrabf ( ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) ∈ { 𝑥𝐴𝑦 ∈ ( 𝐵𝐶 ) } ↔ ( ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) ∈ 𝐴𝑦 ∈ ( ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) / 𝑥 𝐵𝐶 ) ) )
106 105 simplbi ( ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) ∈ { 𝑥𝐴𝑦 ∈ ( 𝐵𝐶 ) } → ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) ∈ 𝐴 )
107 105 simprbi ( ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) ∈ { 𝑥𝐴𝑦 ∈ ( 𝐵𝐶 ) } → 𝑦 ∈ ( ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) / 𝑥 𝐵𝐶 ) )
108 107 ne0d ( ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) ∈ { 𝑥𝐴𝑦 ∈ ( 𝐵𝐶 ) } → ( ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) / 𝑥 𝐵𝐶 ) ≠ ∅ )
109 nfcv 𝑥
110 100 109 nfne 𝑥 ( ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) / 𝑥 𝐵𝐶 ) ≠ ∅
111 103 neeq1d ( 𝑥 = ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) → ( ( 𝐵𝐶 ) ≠ ∅ ↔ ( ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) / 𝑥 𝐵𝐶 ) ≠ ∅ ) )
112 98 58 110 111 elrabf ( ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) ∈ { 𝑥𝐴 ∣ ( 𝐵𝐶 ) ≠ ∅ } ↔ ( ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) ∈ 𝐴 ∧ ( ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) / 𝑥 𝐵𝐶 ) ≠ ∅ ) )
113 106 108 112 sylanbrc ( ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) ∈ { 𝑥𝐴𝑦 ∈ ( 𝐵𝐶 ) } → ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) ∈ { 𝑥𝐴 ∣ ( 𝐵𝐶 ) ≠ ∅ } )
114 96 97 113 3syl ( ( 𝜑𝑦 ∈ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) ) → ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) ∈ { 𝑥𝐴 ∣ ( 𝐵𝐶 ) ≠ ∅ } )
115 114 ralrimiva ( 𝜑 → ∀ 𝑦 ∈ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) ∈ { 𝑥𝐴 ∣ ( 𝐵𝐶 ) ≠ ∅ } )
116 62 38 nfin 𝑥 ( 𝑤 / 𝑥 𝐵𝐶 )
117 116 109 nfne 𝑥 ( 𝑤 / 𝑥 𝐵𝐶 ) ≠ ∅
118 csbeq1a ( 𝑥 = 𝑤𝐵 = 𝑤 / 𝑥 𝐵 )
119 118 ineq1d ( 𝑥 = 𝑤 → ( 𝐵𝐶 ) = ( 𝑤 / 𝑥 𝐵𝐶 ) )
120 119 neeq1d ( 𝑥 = 𝑤 → ( ( 𝐵𝐶 ) ≠ ∅ ↔ ( 𝑤 / 𝑥 𝐵𝐶 ) ≠ ∅ ) )
121 61 58 117 120 elrabf ( 𝑤 ∈ { 𝑥𝐴 ∣ ( 𝐵𝐶 ) ≠ ∅ } ↔ ( 𝑤𝐴 ∧ ( 𝑤 / 𝑥 𝐵𝐶 ) ≠ ∅ ) )
122 121 simprbi ( 𝑤 ∈ { 𝑥𝐴 ∣ ( 𝐵𝐶 ) ≠ ∅ } → ( 𝑤 / 𝑥 𝐵𝐶 ) ≠ ∅ )
123 n0 ( ( 𝑤 / 𝑥 𝐵𝐶 ) ≠ ∅ ↔ ∃ 𝑦 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) )
124 122 123 sylib ( 𝑤 ∈ { 𝑥𝐴 ∣ ( 𝐵𝐶 ) ≠ ∅ } → ∃ 𝑦 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) )
125 124 adantl ( ( 𝜑𝑤 ∈ { 𝑥𝐴 ∣ ( 𝐵𝐶 ) ≠ ∅ } ) → ∃ 𝑦 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) )
126 121 simplbi ( 𝑤 ∈ { 𝑥𝐴 ∣ ( 𝐵𝐶 ) ≠ ∅ } → 𝑤𝐴 )
127 elinel1 ( 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) → 𝑦 𝑤 / 𝑥 𝐵 )
128 127 adantl ( ( ( 𝜑𝑤𝐴 ) ∧ 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) → 𝑦 𝑤 / 𝑥 𝐵 )
129 simplr ( ( ( 𝜑𝑤𝐴 ) ∧ 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) → 𝑤𝐴 )
130 nfv 𝑥 ( 𝜑𝑤𝐴 )
131 62 nfel1 𝑥 𝑤 / 𝑥 𝐵𝑉
132 130 131 nfim 𝑥 ( ( 𝜑𝑤𝐴 ) → 𝑤 / 𝑥 𝐵𝑉 )
133 eleq1w ( 𝑥 = 𝑤 → ( 𝑥𝐴𝑤𝐴 ) )
134 133 anbi2d ( 𝑥 = 𝑤 → ( ( 𝜑𝑥𝐴 ) ↔ ( 𝜑𝑤𝐴 ) ) )
135 118 eleq1d ( 𝑥 = 𝑤 → ( 𝐵𝑉 𝑤 / 𝑥 𝐵𝑉 ) )
136 134 135 imbi12d ( 𝑥 = 𝑤 → ( ( ( 𝜑𝑥𝐴 ) → 𝐵𝑉 ) ↔ ( ( 𝜑𝑤𝐴 ) → 𝑤 / 𝑥 𝐵𝑉 ) ) )
137 132 136 1 chvarfv ( ( 𝜑𝑤𝐴 ) → 𝑤 / 𝑥 𝐵𝑉 )
138 137 adantr ( ( ( 𝜑𝑤𝐴 ) ∧ 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) → 𝑤 / 𝑥 𝐵𝑉 )
139 eqid ( 𝑤𝐴 𝑤 / 𝑥 𝐵 ) = ( 𝑤𝐴 𝑤 / 𝑥 𝐵 )
140 139 elrnmpt1 ( ( 𝑤𝐴 𝑤 / 𝑥 𝐵𝑉 ) → 𝑤 / 𝑥 𝐵 ∈ ran ( 𝑤𝐴 𝑤 / 𝑥 𝐵 ) )
141 129 138 140 syl2anc ( ( ( 𝜑𝑤𝐴 ) ∧ 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) → 𝑤 / 𝑥 𝐵 ∈ ran ( 𝑤𝐴 𝑤 / 𝑥 𝐵 ) )
142 nfcv 𝑤 𝐵
143 118 equcoms ( 𝑤 = 𝑥𝐵 = 𝑤 / 𝑥 𝐵 )
144 143 eqcomd ( 𝑤 = 𝑥 𝑤 / 𝑥 𝐵 = 𝐵 )
145 62 142 144 cbvmpt ( 𝑤𝐴 𝑤 / 𝑥 𝐵 ) = ( 𝑥𝐴𝐵 )
146 145 rneqi ran ( 𝑤𝐴 𝑤 / 𝑥 𝐵 ) = ran ( 𝑥𝐴𝐵 )
147 141 146 eleqtrdi ( ( ( 𝜑𝑤𝐴 ) ∧ 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) → 𝑤 / 𝑥 𝐵 ∈ ran ( 𝑥𝐴𝐵 ) )
148 elunii ( ( 𝑦 𝑤 / 𝑥 𝐵 𝑤 / 𝑥 𝐵 ∈ ran ( 𝑥𝐴𝐵 ) ) → 𝑦 ran ( 𝑥𝐴𝐵 ) )
149 128 147 148 syl2anc ( ( ( 𝜑𝑤𝐴 ) ∧ 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) → 𝑦 ran ( 𝑥𝐴𝐵 ) )
150 elinel2 ( 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) → 𝑦𝐶 )
151 150 adantl ( ( ( 𝜑𝑤𝐴 ) ∧ 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) → 𝑦𝐶 )
152 149 151 elind ( ( ( 𝜑𝑤𝐴 ) ∧ 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) → 𝑦 ∈ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) )
153 nfv 𝑤 𝑦 ∈ ( 𝐵𝐶 )
154 116 nfcri 𝑥 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 )
155 119 eleq2d ( 𝑥 = 𝑤 → ( 𝑦 ∈ ( 𝐵𝐶 ) ↔ 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) )
156 153 154 155 cbvriotaw ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) = ( 𝑤𝐴 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) )
157 simpr ( ( ( 𝜑𝑤𝐴 ) ∧ 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) → 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) )
158 rspe ( ( 𝑤𝐴𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) → ∃ 𝑤𝐴 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) )
159 158 adantll ( ( ( 𝜑𝑤𝐴 ) ∧ 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) → ∃ 𝑤𝐴 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) )
160 simpll ( ( ( 𝜑𝑤𝐴 ) ∧ 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) → 𝜑 )
161 sbequ ( 𝑤 = 𝑧 → ( [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵𝐶 ) ↔ [ 𝑧 / 𝑥 ] 𝑦 ∈ ( 𝐵𝐶 ) ) )
162 sbsbc ( [ 𝑧 / 𝑥 ] 𝑦 ∈ ( 𝐵𝐶 ) ↔ [ 𝑧 / 𝑥 ] 𝑦 ∈ ( 𝐵𝐶 ) )
163 162 a1i ( 𝑤 = 𝑧 → ( [ 𝑧 / 𝑥 ] 𝑦 ∈ ( 𝐵𝐶 ) ↔ [ 𝑧 / 𝑥 ] 𝑦 ∈ ( 𝐵𝐶 ) ) )
164 sbcel2 ( [ 𝑧 / 𝑥 ] 𝑦 ∈ ( 𝐵𝐶 ) ↔ 𝑦 𝑧 / 𝑥 ( 𝐵𝐶 ) )
165 csbin 𝑧 / 𝑥 ( 𝐵𝐶 ) = ( 𝑧 / 𝑥 𝐵 𝑧 / 𝑥 𝐶 )
166 csbconstg ( 𝑧 ∈ V → 𝑧 / 𝑥 𝐶 = 𝐶 )
167 166 elv 𝑧 / 𝑥 𝐶 = 𝐶
168 167 ineq2i ( 𝑧 / 𝑥 𝐵 𝑧 / 𝑥 𝐶 ) = ( 𝑧 / 𝑥 𝐵𝐶 )
169 165 168 eqtri 𝑧 / 𝑥 ( 𝐵𝐶 ) = ( 𝑧 / 𝑥 𝐵𝐶 )
170 169 eleq2i ( 𝑦 𝑧 / 𝑥 ( 𝐵𝐶 ) ↔ 𝑦 ∈ ( 𝑧 / 𝑥 𝐵𝐶 ) )
171 164 170 bitri ( [ 𝑧 / 𝑥 ] 𝑦 ∈ ( 𝐵𝐶 ) ↔ 𝑦 ∈ ( 𝑧 / 𝑥 𝐵𝐶 ) )
172 171 a1i ( 𝑤 = 𝑧 → ( [ 𝑧 / 𝑥 ] 𝑦 ∈ ( 𝐵𝐶 ) ↔ 𝑦 ∈ ( 𝑧 / 𝑥 𝐵𝐶 ) ) )
173 161 163 172 3bitrd ( 𝑤 = 𝑧 → ( [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵𝐶 ) ↔ 𝑦 ∈ ( 𝑧 / 𝑥 𝐵𝐶 ) ) )
174 173 anbi2d ( 𝑤 = 𝑧 → ( ( 𝑦 ∈ ( 𝐵𝐶 ) ∧ [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵𝐶 ) ) ↔ ( 𝑦 ∈ ( 𝐵𝐶 ) ∧ 𝑦 ∈ ( 𝑧 / 𝑥 𝐵𝐶 ) ) ) )
175 equequ2 ( 𝑤 = 𝑧 → ( 𝑥 = 𝑤𝑥 = 𝑧 ) )
176 174 175 imbi12d ( 𝑤 = 𝑧 → ( ( ( 𝑦 ∈ ( 𝐵𝐶 ) ∧ [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵𝐶 ) ) → 𝑥 = 𝑤 ) ↔ ( ( 𝑦 ∈ ( 𝐵𝐶 ) ∧ 𝑦 ∈ ( 𝑧 / 𝑥 𝐵𝐶 ) ) → 𝑥 = 𝑧 ) ) )
177 176 cbvralvw ( ∀ 𝑤𝐴 ( ( 𝑦 ∈ ( 𝐵𝐶 ) ∧ [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵𝐶 ) ) → 𝑥 = 𝑤 ) ↔ ∀ 𝑧𝐴 ( ( 𝑦 ∈ ( 𝐵𝐶 ) ∧ 𝑦 ∈ ( 𝑧 / 𝑥 𝐵𝐶 ) ) → 𝑥 = 𝑧 ) )
178 177 ralbii ( ∀ 𝑥𝐴𝑤𝐴 ( ( 𝑦 ∈ ( 𝐵𝐶 ) ∧ [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵𝐶 ) ) → 𝑥 = 𝑤 ) ↔ ∀ 𝑥𝐴𝑧𝐴 ( ( 𝑦 ∈ ( 𝐵𝐶 ) ∧ 𝑦 ∈ ( 𝑧 / 𝑥 𝐵𝐶 ) ) → 𝑥 = 𝑧 ) )
179 nfv 𝑤𝑧𝐴 ( ( 𝑦 ∈ ( 𝐵𝐶 ) ∧ 𝑦 ∈ ( 𝑧 / 𝑥 𝐵𝐶 ) ) → 𝑥 = 𝑧 )
180 60 38 nfin 𝑥 ( 𝑧 / 𝑥 𝐵𝐶 )
181 180 nfcri 𝑥 𝑦 ∈ ( 𝑧 / 𝑥 𝐵𝐶 )
182 154 181 nfan 𝑥 ( 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ∧ 𝑦 ∈ ( 𝑧 / 𝑥 𝐵𝐶 ) )
183 nfv 𝑥 𝑤 = 𝑧
184 182 183 nfim 𝑥 ( ( 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ∧ 𝑦 ∈ ( 𝑧 / 𝑥 𝐵𝐶 ) ) → 𝑤 = 𝑧 )
185 58 184 nfralw 𝑥𝑧𝐴 ( ( 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ∧ 𝑦 ∈ ( 𝑧 / 𝑥 𝐵𝐶 ) ) → 𝑤 = 𝑧 )
186 155 anbi1d ( 𝑥 = 𝑤 → ( ( 𝑦 ∈ ( 𝐵𝐶 ) ∧ 𝑦 ∈ ( 𝑧 / 𝑥 𝐵𝐶 ) ) ↔ ( 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ∧ 𝑦 ∈ ( 𝑧 / 𝑥 𝐵𝐶 ) ) ) )
187 equequ1 ( 𝑥 = 𝑤 → ( 𝑥 = 𝑧𝑤 = 𝑧 ) )
188 186 187 imbi12d ( 𝑥 = 𝑤 → ( ( ( 𝑦 ∈ ( 𝐵𝐶 ) ∧ 𝑦 ∈ ( 𝑧 / 𝑥 𝐵𝐶 ) ) → 𝑥 = 𝑧 ) ↔ ( ( 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ∧ 𝑦 ∈ ( 𝑧 / 𝑥 𝐵𝐶 ) ) → 𝑤 = 𝑧 ) ) )
189 188 ralbidv ( 𝑥 = 𝑤 → ( ∀ 𝑧𝐴 ( ( 𝑦 ∈ ( 𝐵𝐶 ) ∧ 𝑦 ∈ ( 𝑧 / 𝑥 𝐵𝐶 ) ) → 𝑥 = 𝑧 ) ↔ ∀ 𝑧𝐴 ( ( 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ∧ 𝑦 ∈ ( 𝑧 / 𝑥 𝐵𝐶 ) ) → 𝑤 = 𝑧 ) ) )
190 179 185 189 cbvralw ( ∀ 𝑥𝐴𝑧𝐴 ( ( 𝑦 ∈ ( 𝐵𝐶 ) ∧ 𝑦 ∈ ( 𝑧 / 𝑥 𝐵𝐶 ) ) → 𝑥 = 𝑧 ) ↔ ∀ 𝑤𝐴𝑧𝐴 ( ( 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ∧ 𝑦 ∈ ( 𝑧 / 𝑥 𝐵𝐶 ) ) → 𝑤 = 𝑧 ) )
191 sbsbc ( [ 𝑧 / 𝑤 ] 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ↔ [ 𝑧 / 𝑤 ] 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) )
192 sbcel2 ( [ 𝑧 / 𝑤 ] 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ↔ 𝑦 𝑧 / 𝑤 ( 𝑤 / 𝑥 𝐵𝐶 ) )
193 csbin 𝑧 / 𝑤 ( 𝑤 / 𝑥 𝐵𝐶 ) = ( 𝑧 / 𝑤 𝑤 / 𝑥 𝐵 𝑧 / 𝑤 𝐶 )
194 csbcow 𝑧 / 𝑤 𝑤 / 𝑥 𝐵 = 𝑧 / 𝑥 𝐵
195 csbconstg ( 𝑧 ∈ V → 𝑧 / 𝑤 𝐶 = 𝐶 )
196 195 elv 𝑧 / 𝑤 𝐶 = 𝐶
197 194 196 ineq12i ( 𝑧 / 𝑤 𝑤 / 𝑥 𝐵 𝑧 / 𝑤 𝐶 ) = ( 𝑧 / 𝑥 𝐵𝐶 )
198 193 197 eqtri 𝑧 / 𝑤 ( 𝑤 / 𝑥 𝐵𝐶 ) = ( 𝑧 / 𝑥 𝐵𝐶 )
199 198 eleq2i ( 𝑦 𝑧 / 𝑤 ( 𝑤 / 𝑥 𝐵𝐶 ) ↔ 𝑦 ∈ ( 𝑧 / 𝑥 𝐵𝐶 ) )
200 191 192 199 3bitrri ( 𝑦 ∈ ( 𝑧 / 𝑥 𝐵𝐶 ) ↔ [ 𝑧 / 𝑤 ] 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) )
201 200 anbi2i ( ( 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ∧ 𝑦 ∈ ( 𝑧 / 𝑥 𝐵𝐶 ) ) ↔ ( 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ∧ [ 𝑧 / 𝑤 ] 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) )
202 201 imbi1i ( ( ( 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ∧ 𝑦 ∈ ( 𝑧 / 𝑥 𝐵𝐶 ) ) → 𝑤 = 𝑧 ) ↔ ( ( 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ∧ [ 𝑧 / 𝑤 ] 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) → 𝑤 = 𝑧 ) )
203 202 2ralbii ( ∀ 𝑤𝐴𝑧𝐴 ( ( 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ∧ 𝑦 ∈ ( 𝑧 / 𝑥 𝐵𝐶 ) ) → 𝑤 = 𝑧 ) ↔ ∀ 𝑤𝐴𝑧𝐴 ( ( 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ∧ [ 𝑧 / 𝑤 ] 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) → 𝑤 = 𝑧 ) )
204 178 190 203 3bitri ( ∀ 𝑥𝐴𝑤𝐴 ( ( 𝑦 ∈ ( 𝐵𝐶 ) ∧ [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵𝐶 ) ) → 𝑥 = 𝑤 ) ↔ ∀ 𝑤𝐴𝑧𝐴 ( ( 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ∧ [ 𝑧 / 𝑤 ] 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) → 𝑤 = 𝑧 ) )
205 94 204 sylib ( ( 𝜑𝑦 ∈ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) ) → ∀ 𝑤𝐴𝑧𝐴 ( ( 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ∧ [ 𝑧 / 𝑤 ] 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) → 𝑤 = 𝑧 ) )
206 160 152 205 syl2anc ( ( ( 𝜑𝑤𝐴 ) ∧ 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) → ∀ 𝑤𝐴𝑧𝐴 ( ( 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ∧ [ 𝑧 / 𝑤 ] 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) → 𝑤 = 𝑧 ) )
207 reu2 ( ∃! 𝑤𝐴 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ↔ ( ∃ 𝑤𝐴 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ∧ ∀ 𝑤𝐴𝑧𝐴 ( ( 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ∧ [ 𝑧 / 𝑤 ] 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) → 𝑤 = 𝑧 ) ) )
208 159 206 207 sylanbrc ( ( ( 𝜑𝑤𝐴 ) ∧ 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) → ∃! 𝑤𝐴 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) )
209 riota1 ( ∃! 𝑤𝐴 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) → ( ( 𝑤𝐴𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) ↔ ( 𝑤𝐴 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) = 𝑤 ) )
210 208 209 syl ( ( ( 𝜑𝑤𝐴 ) ∧ 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) → ( ( 𝑤𝐴𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) ↔ ( 𝑤𝐴 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) = 𝑤 ) )
211 129 157 210 mpbi2and ( ( ( 𝜑𝑤𝐴 ) ∧ 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) → ( 𝑤𝐴 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) = 𝑤 )
212 156 211 eqtr2id ( ( ( 𝜑𝑤𝐴 ) ∧ 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) → 𝑤 = ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) )
213 152 212 jca ( ( ( 𝜑𝑤𝐴 ) ∧ 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) ) → ( 𝑦 ∈ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) ∧ 𝑤 = ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) ) )
214 213 ex ( ( 𝜑𝑤𝐴 ) → ( 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) → ( 𝑦 ∈ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) ∧ 𝑤 = ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) ) ) )
215 126 214 sylan2 ( ( 𝜑𝑤 ∈ { 𝑥𝐴 ∣ ( 𝐵𝐶 ) ≠ ∅ } ) → ( 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) → ( 𝑦 ∈ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) ∧ 𝑤 = ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) ) ) )
216 215 eximdv ( ( 𝜑𝑤 ∈ { 𝑥𝐴 ∣ ( 𝐵𝐶 ) ≠ ∅ } ) → ( ∃ 𝑦 𝑦 ∈ ( 𝑤 / 𝑥 𝐵𝐶 ) → ∃ 𝑦 ( 𝑦 ∈ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) ∧ 𝑤 = ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) ) ) )
217 125 216 mpd ( ( 𝜑𝑤 ∈ { 𝑥𝐴 ∣ ( 𝐵𝐶 ) ≠ ∅ } ) → ∃ 𝑦 ( 𝑦 ∈ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) ∧ 𝑤 = ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) ) )
218 df-rex ( ∃ 𝑦 ∈ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) 𝑤 = ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) ↔ ∃ 𝑦 ( 𝑦 ∈ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) ∧ 𝑤 = ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) ) )
219 217 218 sylibr ( ( 𝜑𝑤 ∈ { 𝑥𝐴 ∣ ( 𝐵𝐶 ) ≠ ∅ } ) → ∃ 𝑦 ∈ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) 𝑤 = ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) )
220 219 ralrimiva ( 𝜑 → ∀ 𝑤 ∈ { 𝑥𝐴 ∣ ( 𝐵𝐶 ) ≠ ∅ } ∃ 𝑦 ∈ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) 𝑤 = ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) )
221 eqid ( 𝑦 ∈ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) ↦ ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) ) = ( 𝑦 ∈ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) ↦ ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) )
222 221 fompt ( ( 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 ) ) ) )
223 115 220 222 sylanbrc ( 𝜑 → ( 𝑦 ∈ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) ↦ ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) ) : ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) –onto→ { 𝑥𝐴 ∣ ( 𝐵𝐶 ) ≠ ∅ } )
224 fodomg ( ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) ∈ V → ( ( 𝑦 ∈ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) ↦ ( 𝑥𝐴 𝑦 ∈ ( 𝐵𝐶 ) ) ) : ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) –onto→ { 𝑥𝐴 ∣ ( 𝐵𝐶 ) ≠ ∅ } → { 𝑥𝐴 ∣ ( 𝐵𝐶 ) ≠ ∅ } ≼ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) ) )
225 8 223 224 sylc ( 𝜑 → { 𝑥𝐴 ∣ ( 𝐵𝐶 ) ≠ ∅ } ≼ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) )
226 domfi ( ( ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) ∈ Fin ∧ { 𝑥𝐴 ∣ ( 𝐵𝐶 ) ≠ ∅ } ≼ ( ran ( 𝑥𝐴𝐵 ) ∩ 𝐶 ) ) → { 𝑥𝐴 ∣ ( 𝐵𝐶 ) ≠ ∅ } ∈ Fin )
227 6 225 226 syl2anc ( 𝜑 → { 𝑥𝐴 ∣ ( 𝐵𝐶 ) ≠ ∅ } ∈ Fin )