Metamath Proof Explorer


Theorem fnwe2lem2

Description: Lemma for fnwe2 . An element which is in a minimal fiber and minimal within its fiber is minimal globally; thus T is well-founded. (Contributed by Stefan O'Rear, 19-Jan-2015)

Ref Expression
Hypotheses fnwe2.su z = F x S = U
fnwe2.t T = x y | F x R F y F x = F y x U y
fnwe2.s φ x A U We y A | F y = F x
fnwe2.f φ F A : A B
fnwe2.r φ R We B
fnwe2lem2.a φ a A
fnwe2lem2.n0 φ a
Assertion fnwe2lem2 φ b a c a ¬ c T b

Proof

Step Hyp Ref Expression
1 fnwe2.su z = F x S = U
2 fnwe2.t T = x y | F x R F y F x = F y x U y
3 fnwe2.s φ x A U We y A | F y = F x
4 fnwe2.f φ F A : A B
5 fnwe2.r φ R We B
6 fnwe2lem2.a φ a A
7 fnwe2lem2.n0 φ a
8 ffun F A : A B Fun F A
9 vex a V
10 9 funimaex Fun F A F A a V
11 4 8 10 3syl φ F A a V
12 wefr R We B R Fr B
13 5 12 syl φ R Fr B
14 imassrn F A a ran F A
15 4 frnd φ ran F A B
16 14 15 sstrid φ F A a B
17 incom dom F A a = a dom F A
18 4 fdmd φ dom F A = A
19 6 18 sseqtrrd φ a dom F A
20 df-ss a dom F A a dom F A = a
21 19 20 sylib φ a dom F A = a
22 17 21 syl5eq φ dom F A a = a
23 22 7 eqnetrd φ dom F A a
24 imadisj F A a = dom F A a =
25 24 necon3bii F A a dom F A a
26 23 25 sylibr φ F A a
27 fri F A a V R Fr B F A a B F A a d F A a e F A a ¬ e R d
28 11 13 16 26 27 syl22anc φ d F A a e F A a ¬ e R d
29 df-ima F A a = ran F A a
30 29 rexeqi d F A a e F A a ¬ e R d d ran F A a e F A a ¬ e R d
31 4 ffnd φ F A Fn A
32 fnssres F A Fn A a A F A a Fn a
33 31 6 32 syl2anc φ F A a Fn a
34 breq2 d = F A a f e R d e R F A a f
35 34 notbid d = F A a f ¬ e R d ¬ e R F A a f
36 35 ralbidv d = F A a f e F A a ¬ e R d e F A a ¬ e R F A a f
37 36 rexrn F A a Fn a d ran F A a e F A a ¬ e R d f a e F A a ¬ e R F A a f
38 33 37 syl φ d ran F A a e F A a ¬ e R d f a e F A a ¬ e R F A a f
39 30 38 syl5bb φ d F A a e F A a ¬ e R d f a e F A a ¬ e R F A a f
40 29 raleqi e F A a ¬ e R F A a f e ran F A a ¬ e R F A a f
41 breq1 e = F A a d e R F A a f F A a d R F A a f
42 41 notbid e = F A a d ¬ e R F A a f ¬ F A a d R F A a f
43 42 ralrn F A a Fn a e ran F A a ¬ e R F A a f d a ¬ F A a d R F A a f
44 33 43 syl φ e ran F A a ¬ e R F A a f d a ¬ F A a d R F A a f
45 40 44 syl5bb φ e F A a ¬ e R F A a f d a ¬ F A a d R F A a f
46 45 adantr φ f a e F A a ¬ e R F A a f d a ¬ F A a d R F A a f
47 6 resabs1d φ F A a = F a
48 47 ad2antrr φ f a d a F A a = F a
49 48 fveq1d φ f a d a F A a d = F a d
50 fvres d a F a d = F d
51 50 adantl φ f a d a F a d = F d
52 49 51 eqtrd φ f a d a F A a d = F d
53 48 fveq1d φ f a d a F A a f = F a f
54 fvres f a F a f = F f
55 54 ad2antlr φ f a d a F a f = F f
56 53 55 eqtrd φ f a d a F A a f = F f
57 52 56 breq12d φ f a d a F A a d R F A a f F d R F f
58 57 notbid φ f a d a ¬ F A a d R F A a f ¬ F d R F f
59 58 ralbidva φ f a d a ¬ F A a d R F A a f d a ¬ F d R F f
60 46 59 bitrd φ f a e F A a ¬ e R F A a f d a ¬ F d R F f
61 60 rexbidva φ f a e F A a ¬ e R F A a f f a d a ¬ F d R F f
62 39 61 bitrd φ d F A a e F A a ¬ e R d f a d a ¬ F d R F f
63 9 inex1 a y A | F y = F f V
64 63 a1i φ f a d a ¬ F d R F f a y A | F y = F f V
65 6 sselda φ f a f A
66 1 2 3 fnwe2lem1 φ f A F f / z S We y A | F y = F f
67 wefr F f / z S We y A | F y = F f F f / z S Fr y A | F y = F f
68 66 67 syl φ f A F f / z S Fr y A | F y = F f
69 65 68 syldan φ f a F f / z S Fr y A | F y = F f
70 69 adantrr φ f a d a ¬ F d R F f F f / z S Fr y A | F y = F f
71 inss2 a y A | F y = F f y A | F y = F f
72 71 a1i φ f a d a ¬ F d R F f a y A | F y = F f y A | F y = F f
73 simprl φ f a d a ¬ F d R F f f a
74 fveqeq2 y = f F y = F f F f = F f
75 65 adantrr φ f a d a ¬ F d R F f f A
76 eqidd φ f a d a ¬ F d R F f F f = F f
77 74 75 76 elrabd φ f a d a ¬ F d R F f f y A | F y = F f
78 73 77 elind φ f a d a ¬ F d R F f f a y A | F y = F f
79 78 ne0d φ f a d a ¬ F d R F f a y A | F y = F f
80 fri a y A | F y = F f V F f / z S Fr y A | F y = F f a y A | F y = F f y A | F y = F f a y A | F y = F f e a y A | F y = F f g a y A | F y = F f ¬ g F f / z S e
81 64 70 72 79 80 syl22anc φ f a d a ¬ F d R F f e a y A | F y = F f g a y A | F y = F f ¬ g F f / z S e
82 elin e a y A | F y = F f e a e y A | F y = F f
83 fveqeq2 y = e F y = F f F e = F f
84 83 elrab e y A | F y = F f e A F e = F f
85 84 anbi2i e a e y A | F y = F f e a e A F e = F f
86 82 85 bitri e a y A | F y = F f e a e A F e = F f
87 elin g a y A | F y = F f g a g y A | F y = F f
88 fveqeq2 y = g F y = F f F g = F f
89 88 elrab g y A | F y = F f g A F g = F f
90 89 anbi2i g a g y A | F y = F f g a g A F g = F f
91 87 90 bitri g a y A | F y = F f g a g A F g = F f
92 91 imbi1i g a y A | F y = F f ¬ g F f / z S e g a g A F g = F f ¬ g F f / z S e
93 impexp g a g A F g = F f ¬ g F f / z S e g a g A F g = F f ¬ g F f / z S e
94 92 93 bitri g a y A | F y = F f ¬ g F f / z S e g a g A F g = F f ¬ g F f / z S e
95 94 ralbii2 g a y A | F y = F f ¬ g F f / z S e g a g A F g = F f ¬ g F f / z S e
96 simplrl φ f a d a ¬ F d R F f e a e A F e = F f g a g A F g = F f ¬ g F f / z S e e a
97 fveq2 d = c F d = F c
98 97 breq1d d = c F d R F f F c R F f
99 98 notbid d = c ¬ F d R F f ¬ F c R F f
100 simplrr φ f a d a ¬ F d R F f e a e A F e = F f d a ¬ F d R F f
101 100 ad2antrr φ f a d a ¬ F d R F f e a e A F e = F f g a g A F g = F f ¬ g F f / z S e c a d a ¬ F d R F f
102 simpr φ f a d a ¬ F d R F f e a e A F e = F f g a g A F g = F f ¬ g F f / z S e c a c a
103 99 101 102 rspcdva φ f a d a ¬ F d R F f e a e A F e = F f g a g A F g = F f ¬ g F f / z S e c a ¬ F c R F f
104 simprrr φ f a d a ¬ F d R F f e a e A F e = F f F e = F f
105 104 ad2antrr φ f a d a ¬ F d R F f e a e A F e = F f g a g A F g = F f ¬ g F f / z S e c a F e = F f
106 105 breq2d φ f a d a ¬ F d R F f e a e A F e = F f g a g A F g = F f ¬ g F f / z S e c a F c R F e F c R F f
107 103 106 mtbird φ f a d a ¬ F d R F f e a e A F e = F f g a g A F g = F f ¬ g F f / z S e c a ¬ F c R F e
108 6 ad3antrrr φ f a d a ¬ F d R F f e a e A F e = F f g a g A F g = F f ¬ g F f / z S e a A
109 108 sselda φ f a d a ¬ F d R F f e a e A F e = F f g a g A F g = F f ¬ g F f / z S e c a c A
110 109 adantrr φ f a d a ¬ F d R F f e a e A F e = F f g a g A F g = F f ¬ g F f / z S e c a F c = F e c A
111 simprr φ f a d a ¬ F d R F f e a e A F e = F f g a g A F g = F f ¬ g F f / z S e c a F c = F e F c = F e
112 104 ad2antrr φ f a d a ¬ F d R F f e a e A F e = F f g a g A F g = F f ¬ g F f / z S e c a F c = F e F e = F f
113 111 112 eqtrd φ f a d a ¬ F d R F f e a e A F e = F f g a g A F g = F f ¬ g F f / z S e c a F c = F e F c = F f
114 eleq1w g = c g A c A
115 fveqeq2 g = c F g = F f F c = F f
116 114 115 anbi12d g = c g A F g = F f c A F c = F f
117 breq1 g = c g F f / z S e c F f / z S e
118 117 notbid g = c ¬ g F f / z S e ¬ c F f / z S e
119 116 118 imbi12d g = c g A F g = F f ¬ g F f / z S e c A F c = F f ¬ c F f / z S e
120 simplr φ f a d a ¬ F d R F f e a e A F e = F f g a g A F g = F f ¬ g F f / z S e c a F c = F e g a g A F g = F f ¬ g F f / z S e
121 simprl φ f a d a ¬ F d R F f e a e A F e = F f g a g A F g = F f ¬ g F f / z S e c a F c = F e c a
122 119 120 121 rspcdva φ f a d a ¬ F d R F f e a e A F e = F f g a g A F g = F f ¬ g F f / z S e c a F c = F e c A F c = F f ¬ c F f / z S e
123 110 113 122 mp2and φ f a d a ¬ F d R F f e a e A F e = F f g a g A F g = F f ¬ g F f / z S e c a F c = F e ¬ c F f / z S e
124 111 112 eqtr2d φ f a d a ¬ F d R F f e a e A F e = F f g a g A F g = F f ¬ g F f / z S e c a F c = F e F f = F c
125 124 csbeq1d φ f a d a ¬ F d R F f e a e A F e = F f g a g A F g = F f ¬ g F f / z S e c a F c = F e F f / z S = F c / z S
126 125 breqd φ f a d a ¬ F d R F f e a e A F e = F f g a g A F g = F f ¬ g F f / z S e c a F c = F e c F f / z S e c F c / z S e
127 123 126 mtbid φ f a d a ¬ F d R F f e a e A F e = F f g a g A F g = F f ¬ g F f / z S e c a F c = F e ¬ c F c / z S e
128 127 expr φ f a d a ¬ F d R F f e a e A F e = F f g a g A F g = F f ¬ g F f / z S e c a F c = F e ¬ c F c / z S e
129 imnan F c = F e ¬ c F c / z S e ¬ F c = F e c F c / z S e
130 128 129 sylib φ f a d a ¬ F d R F f e a e A F e = F f g a g A F g = F f ¬ g F f / z S e c a ¬ F c = F e c F c / z S e
131 ioran ¬ F c R F e F c = F e c F c / z S e ¬ F c R F e ¬ F c = F e c F c / z S e
132 107 130 131 sylanbrc φ f a d a ¬ F d R F f e a e A F e = F f g a g A F g = F f ¬ g F f / z S e c a ¬ F c R F e F c = F e c F c / z S e
133 1 2 fnwe2val c T e F c R F e F c = F e c F c / z S e
134 132 133 sylnibr φ f a d a ¬ F d R F f e a e A F e = F f g a g A F g = F f ¬ g F f / z S e c a ¬ c T e
135 134 ralrimiva φ f a d a ¬ F d R F f e a e A F e = F f g a g A F g = F f ¬ g F f / z S e c a ¬ c T e
136 breq2 b = e c T b c T e
137 136 notbid b = e ¬ c T b ¬ c T e
138 137 ralbidv b = e c a ¬ c T b c a ¬ c T e
139 138 rspcev e a c a ¬ c T e b a c a ¬ c T b
140 96 135 139 syl2anc φ f a d a ¬ F d R F f e a e A F e = F f g a g A F g = F f ¬ g F f / z S e b a c a ¬ c T b
141 140 ex φ f a d a ¬ F d R F f e a e A F e = F f g a g A F g = F f ¬ g F f / z S e b a c a ¬ c T b
142 95 141 syl5bi φ f a d a ¬ F d R F f e a e A F e = F f g a y A | F y = F f ¬ g F f / z S e b a c a ¬ c T b
143 142 ex φ f a d a ¬ F d R F f e a e A F e = F f g a y A | F y = F f ¬ g F f / z S e b a c a ¬ c T b
144 86 143 syl5bi φ f a d a ¬ F d R F f e a y A | F y = F f g a y A | F y = F f ¬ g F f / z S e b a c a ¬ c T b
145 144 rexlimdv φ f a d a ¬ F d R F f e a y A | F y = F f g a y A | F y = F f ¬ g F f / z S e b a c a ¬ c T b
146 81 145 mpd φ f a d a ¬ F d R F f b a c a ¬ c T b
147 146 rexlimdvaa φ f a d a ¬ F d R F f b a c a ¬ c T b
148 62 147 sylbid φ d F A a e F A a ¬ e R d b a c a ¬ c T b
149 28 148 mpd φ b a c a ¬ c T b