Metamath Proof Explorer


Theorem domunfican

Description: A finite set union cancellation law for dominance. (Contributed by Stefan O'Rear, 19-Feb-2015) (Revised by Stefan O'Rear, 5-May-2015)

Ref Expression
Assertion domunfican A Fin B A A X = B Y = A X B Y X Y

Proof

Step Hyp Ref Expression
1 ensym B A A B
2 bren A B f f : A 1-1 onto B
3 1 2 sylib B A f f : A 1-1 onto B
4 ssid A A
5 sseq1 a = a A A
6 5 anbi1d a = a A f : A 1-1 onto B A f : A 1-1 onto B
7 6 anbi1d a = a A f : A 1-1 onto B A X = B Y = A f : A 1-1 onto B A X = B Y =
8 uneq1 a = a X = X
9 imaeq2 a = f a = f
10 9 uneq1d a = f a Y = f Y
11 8 10 breq12d a = a X f a Y X f Y
12 11 bibi1d a = a X f a Y X Y X f Y X Y
13 7 12 imbi12d a = a A f : A 1-1 onto B A X = B Y = a X f a Y X Y A f : A 1-1 onto B A X = B Y = X f Y X Y
14 sseq1 a = b a A b A
15 14 anbi1d a = b a A f : A 1-1 onto B b A f : A 1-1 onto B
16 15 anbi1d a = b a A f : A 1-1 onto B A X = B Y = b A f : A 1-1 onto B A X = B Y =
17 uneq1 a = b a X = b X
18 imaeq2 a = b f a = f b
19 18 uneq1d a = b f a Y = f b Y
20 17 19 breq12d a = b a X f a Y b X f b Y
21 20 bibi1d a = b a X f a Y X Y b X f b Y X Y
22 16 21 imbi12d a = b a A f : A 1-1 onto B A X = B Y = a X f a Y X Y b A f : A 1-1 onto B A X = B Y = b X f b Y X Y
23 sseq1 a = b c a A b c A
24 23 anbi1d a = b c a A f : A 1-1 onto B b c A f : A 1-1 onto B
25 24 anbi1d a = b c a A f : A 1-1 onto B A X = B Y = b c A f : A 1-1 onto B A X = B Y =
26 uneq1 a = b c a X = b c X
27 imaeq2 a = b c f a = f b c
28 27 uneq1d a = b c f a Y = f b c Y
29 26 28 breq12d a = b c a X f a Y b c X f b c Y
30 29 bibi1d a = b c a X f a Y X Y b c X f b c Y X Y
31 25 30 imbi12d a = b c a A f : A 1-1 onto B A X = B Y = a X f a Y X Y b c A f : A 1-1 onto B A X = B Y = b c X f b c Y X Y
32 sseq1 a = A a A A A
33 32 anbi1d a = A a A f : A 1-1 onto B A A f : A 1-1 onto B
34 33 anbi1d a = A a A f : A 1-1 onto B A X = B Y = A A f : A 1-1 onto B A X = B Y =
35 uneq1 a = A a X = A X
36 imaeq2 a = A f a = f A
37 36 uneq1d a = A f a Y = f A Y
38 35 37 breq12d a = A a X f a Y A X f A Y
39 38 bibi1d a = A a X f a Y X Y A X f A Y X Y
40 34 39 imbi12d a = A a A f : A 1-1 onto B A X = B Y = a X f a Y X Y A A f : A 1-1 onto B A X = B Y = A X f A Y X Y
41 uncom X = X
42 un0 X = X
43 41 42 eqtri X = X
44 ima0 f =
45 44 uneq1i f Y = Y
46 uncom Y = Y
47 un0 Y = Y
48 46 47 eqtri Y = Y
49 45 48 eqtri f Y = Y
50 43 49 breq12i X f Y X Y
51 50 a1i A f : A 1-1 onto B A X = B Y = X f Y X Y
52 ssun1 b b c
53 sstr2 b b c b c A b A
54 52 53 ax-mp b c A b A
55 54 anim1i b c A f : A 1-1 onto B b A f : A 1-1 onto B
56 55 anim1i b c A f : A 1-1 onto B A X = B Y = b A f : A 1-1 onto B A X = B Y =
57 56 imim1i b A f : A 1-1 onto B A X = B Y = b X f b Y X Y b c A f : A 1-1 onto B A X = B Y = b X f b Y X Y
58 uncom b c = c b
59 58 uneq1i b c X = c b X
60 unass c b X = c b X
61 59 60 eqtri b c X = c b X
62 61 a1i b Fin ¬ c b b c A f : A 1-1 onto B A X = B Y = b c X = c b X
63 imaundi f b c = f b f c
64 simprlr b Fin ¬ c b b c A f : A 1-1 onto B A X = B Y = f : A 1-1 onto B
65 f1ofn f : A 1-1 onto B f Fn A
66 64 65 syl b Fin ¬ c b b c A f : A 1-1 onto B A X = B Y = f Fn A
67 ssun2 c b c
68 sstr2 c b c b c A c A
69 67 68 ax-mp b c A c A
70 vex c V
71 70 snss c A c A
72 69 71 sylibr b c A c A
73 72 adantr b c A f : A 1-1 onto B c A
74 73 ad2antrl b Fin ¬ c b b c A f : A 1-1 onto B A X = B Y = c A
75 fnsnfv f Fn A c A f c = f c
76 66 74 75 syl2anc b Fin ¬ c b b c A f : A 1-1 onto B A X = B Y = f c = f c
77 76 eqcomd b Fin ¬ c b b c A f : A 1-1 onto B A X = B Y = f c = f c
78 77 uneq2d b Fin ¬ c b b c A f : A 1-1 onto B A X = B Y = f b f c = f b f c
79 63 78 eqtrid b Fin ¬ c b b c A f : A 1-1 onto B A X = B Y = f b c = f b f c
80 79 uneq1d b Fin ¬ c b b c A f : A 1-1 onto B A X = B Y = f b c Y = f b f c Y
81 uncom f b f c = f c f b
82 81 uneq1i f b f c Y = f c f b Y
83 unass f c f b Y = f c f b Y
84 82 83 eqtri f b f c Y = f c f b Y
85 80 84 eqtrdi b Fin ¬ c b b c A f : A 1-1 onto B A X = B Y = f b c Y = f c f b Y
86 62 85 breq12d b Fin ¬ c b b c A f : A 1-1 onto B A X = B Y = b c X f b c Y c b X f c f b Y
87 simplr b Fin ¬ c b b c A f : A 1-1 onto B A X = B Y = ¬ c b
88 incom X A = A X
89 simprrl b Fin ¬ c b b c A f : A 1-1 onto B A X = B Y = A X =
90 88 89 eqtrid b Fin ¬ c b b c A f : A 1-1 onto B A X = B Y = X A =
91 minel c A X A = ¬ c X
92 74 90 91 syl2anc b Fin ¬ c b b c A f : A 1-1 onto B A X = B Y = ¬ c X
93 ioran ¬ c b c X ¬ c b ¬ c X
94 elun c b X c b c X
95 93 94 xchnxbir ¬ c b X ¬ c b ¬ c X
96 87 92 95 sylanbrc b Fin ¬ c b b c A f : A 1-1 onto B A X = B Y = ¬ c b X
97 simplr b Fin ¬ c b b c A f : A 1-1 onto B ¬ c b
98 f1of1 f : A 1-1 onto B f : A 1-1 B
99 98 adantl b c A f : A 1-1 onto B f : A 1-1 B
100 54 adantr b c A f : A 1-1 onto B b A
101 f1elima f : A 1-1 B c A b A f c f b c b
102 99 73 100 101 syl3anc b c A f : A 1-1 onto B f c f b c b
103 102 biimpd b c A f : A 1-1 onto B f c f b c b
104 103 adantl b Fin ¬ c b b c A f : A 1-1 onto B f c f b c b
105 97 104 mtod b Fin ¬ c b b c A f : A 1-1 onto B ¬ f c f b
106 105 adantrr b Fin ¬ c b b c A f : A 1-1 onto B A X = B Y = ¬ f c f b
107 f1of f : A 1-1 onto B f : A B
108 64 107 syl b Fin ¬ c b b c A f : A 1-1 onto B A X = B Y = f : A B
109 108 74 ffvelrnd b Fin ¬ c b b c A f : A 1-1 onto B A X = B Y = f c B
110 incom Y B = B Y
111 simprrr b Fin ¬ c b b c A f : A 1-1 onto B A X = B Y = B Y =
112 110 111 eqtrid b Fin ¬ c b b c A f : A 1-1 onto B A X = B Y = Y B =
113 minel f c B Y B = ¬ f c Y
114 109 112 113 syl2anc b Fin ¬ c b b c A f : A 1-1 onto B A X = B Y = ¬ f c Y
115 ioran ¬ f c f b f c Y ¬ f c f b ¬ f c Y
116 elun f c f b Y f c f b f c Y
117 115 116 xchnxbir ¬ f c f b Y ¬ f c f b ¬ f c Y
118 106 114 117 sylanbrc b Fin ¬ c b b c A f : A 1-1 onto B A X = B Y = ¬ f c f b Y
119 fvex f c V
120 70 119 domunsncan ¬ c b X ¬ f c f b Y c b X f c f b Y b X f b Y
121 96 118 120 syl2anc b Fin ¬ c b b c A f : A 1-1 onto B A X = B Y = c b X f c f b Y b X f b Y
122 86 121 bitrd b Fin ¬ c b b c A f : A 1-1 onto B A X = B Y = b c X f b c Y b X f b Y
123 bitr b c X f b c Y b X f b Y b X f b Y X Y b c X f b c Y X Y
124 123 ex b c X f b c Y b X f b Y b X f b Y X Y b c X f b c Y X Y
125 122 124 syl b Fin ¬ c b b c A f : A 1-1 onto B A X = B Y = b X f b Y X Y b c X f b c Y X Y
126 125 ex b Fin ¬ c b b c A f : A 1-1 onto B A X = B Y = b X f b Y X Y b c X f b c Y X Y
127 126 a2d b Fin ¬ c b b c A f : A 1-1 onto B A X = B Y = b X f b Y X Y b c A f : A 1-1 onto B A X = B Y = b c X f b c Y X Y
128 57 127 syl5 b Fin ¬ c b b A f : A 1-1 onto B A X = B Y = b X f b Y X Y b c A f : A 1-1 onto B A X = B Y = b c X f b c Y X Y
129 13 22 31 40 51 128 findcard2s A Fin A A f : A 1-1 onto B A X = B Y = A X f A Y X Y
130 129 expd A Fin A A f : A 1-1 onto B A X = B Y = A X f A Y X Y
131 4 130 mpani A Fin f : A 1-1 onto B A X = B Y = A X f A Y X Y
132 131 3imp A Fin f : A 1-1 onto B A X = B Y = A X f A Y X Y
133 f1ofo f : A 1-1 onto B f : A onto B
134 foima f : A onto B f A = B
135 133 134 syl f : A 1-1 onto B f A = B
136 135 uneq1d f : A 1-1 onto B f A Y = B Y
137 136 breq2d f : A 1-1 onto B A X f A Y A X B Y
138 137 bibi1d f : A 1-1 onto B A X f A Y X Y A X B Y X Y
139 138 3ad2ant2 A Fin f : A 1-1 onto B A X = B Y = A X f A Y X Y A X B Y X Y
140 132 139 mpbid A Fin f : A 1-1 onto B A X = B Y = A X B Y X Y
141 140 3exp A Fin f : A 1-1 onto B A X = B Y = A X B Y X Y
142 141 exlimdv A Fin f f : A 1-1 onto B A X = B Y = A X B Y X Y
143 3 142 syl5 A Fin B A A X = B Y = A X B Y X Y
144 143 imp31 A Fin B A A X = B Y = A X B Y X Y