Metamath Proof Explorer


Theorem brapply

Description: Binary relation form of the Apply function. (Contributed by Scott Fenton, 12-Apr-2014) (Revised by Mario Carneiro, 19-Apr-2014) (Proof shortened by Peter Mazsa, 2-Oct-2022)

Ref Expression
Hypotheses brapply.1 A V
brapply.2 B V
brapply.3 C V
Assertion brapply A B 𝖠𝗉𝗉𝗅𝗒 C C = A B

Proof

Step Hyp Ref Expression
1 brapply.1 A V
2 brapply.2 B V
3 brapply.3 C V
4 snex A B V
5 4 inex1 A B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 V
6 unieq x = A B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 x = A B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌
7 6 unieqd x = A B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 x = A B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌
8 7 eqeq2d x = A B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 C = x C = A B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌
9 5 8 ceqsexv x x = A B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 C = x C = A B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌
10 df-apply 𝖠𝗉𝗉𝗅𝗒 = 𝖡𝗂𝗀𝖼𝗎𝗉 𝖡𝗂𝗀𝖼𝗎𝗉 V × V ran V E E 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 V 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 𝖨𝗆𝗀 pprod I 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇
11 10 breqi A B 𝖠𝗉𝗉𝗅𝗒 C A B 𝖡𝗂𝗀𝖼𝗎𝗉 𝖡𝗂𝗀𝖼𝗎𝗉 V × V ran V E E 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 V 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 𝖨𝗆𝗀 pprod I 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 C
12 opex A B V
13 12 3 brco A B 𝖡𝗂𝗀𝖼𝗎𝗉 𝖡𝗂𝗀𝖼𝗎𝗉 V × V ran V E E 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 V 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 𝖨𝗆𝗀 pprod I 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 C x A B V × V ran V E E 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 V 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 𝖨𝗆𝗀 pprod I 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 x x 𝖡𝗂𝗀𝖼𝗎𝗉 𝖡𝗂𝗀𝖼𝗎𝗉 C
14 vex x V
15 12 14 brco A B V × V ran V E E 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 V 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 𝖨𝗆𝗀 pprod I 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 x y A B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 𝖨𝗆𝗀 pprod I 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 y y V × V ran V E E 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 V x
16 vex y V
17 12 16 brco A B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 𝖨𝗆𝗀 pprod I 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 y z A B pprod I 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 z z 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 𝖨𝗆𝗀 y
18 vex z V
19 1 2 18 brpprod3a A B pprod I 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 z a b z = a b A I a B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 b
20 3anrot z = a b A I a B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 b A I a B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 b z = a b
21 vex a V
22 21 ideq A I a A = a
23 eqcom A = a a = A
24 22 23 bitri A I a a = A
25 vex b V
26 2 25 brsingle B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 b b = B
27 biid z = a b z = a b
28 24 26 27 3anbi123i A I a B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 b z = a b a = A b = B z = a b
29 20 28 bitri z = a b A I a B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 b a = A b = B z = a b
30 29 2exbii a b z = a b A I a B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 b a b a = A b = B z = a b
31 snex B V
32 opeq1 a = A a b = A b
33 32 eqeq2d a = A z = a b z = A b
34 opeq2 b = B A b = A B
35 34 eqeq2d b = B z = A b z = A B
36 1 31 33 35 ceqsex2v a b a = A b = B z = a b z = A B
37 19 30 36 3bitri A B pprod I 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 z z = A B
38 37 anbi1i A B pprod I 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 z z 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 𝖨𝗆𝗀 y z = A B z 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 𝖨𝗆𝗀 y
39 38 exbii z A B pprod I 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 z z 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 𝖨𝗆𝗀 y z z = A B z 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 𝖨𝗆𝗀 y
40 opex A B V
41 breq1 z = A B z 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 𝖨𝗆𝗀 y A B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 𝖨𝗆𝗀 y
42 40 41 ceqsexv z z = A B z 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 𝖨𝗆𝗀 y A B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 𝖨𝗆𝗀 y
43 40 16 brco A B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 𝖨𝗆𝗀 y x A B 𝖨𝗆𝗀 x x 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 y
44 1 31 14 brimg A B 𝖨𝗆𝗀 x x = A B
45 14 16 brsingle x 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 y y = x
46 44 45 anbi12i A B 𝖨𝗆𝗀 x x 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 y x = A B y = x
47 46 exbii x A B 𝖨𝗆𝗀 x x 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 y x x = A B y = x
48 1 imaex A B V
49 sneq x = A B x = A B
50 49 eqeq2d x = A B y = x y = A B
51 48 50 ceqsexv x x = A B y = x y = A B
52 47 51 bitri x A B 𝖨𝗆𝗀 x x 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 y y = A B
53 42 43 52 3bitri z z = A B z 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 𝖨𝗆𝗀 y y = A B
54 17 39 53 3bitri A B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 𝖨𝗆𝗀 pprod I 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 y y = A B
55 eqid V × V ran V E E 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 V = V × V ran V E E 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 V
56 brxp y V × V x y V x V
57 16 14 56 mpbir2an y V × V x
58 epel z E y z y
59 58 anbi1ci z 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 z E y z y z 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌
60 16 brresi z E 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 y z 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 z E y
61 elin z y 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 z y z 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌
62 59 60 61 3bitr4ri z y 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 z E 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 y
63 16 14 55 57 62 brtxpsd3 y V × V ran V E E 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 V x x = y 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌
64 54 63 anbi12i A B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 𝖨𝗆𝗀 pprod I 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 y y V × V ran V E E 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 V x y = A B x = y 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌
65 64 exbii y A B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 𝖨𝗆𝗀 pprod I 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 y y V × V ran V E E 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 V x y y = A B x = y 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌
66 ineq1 y = A B y 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 = A B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌
67 66 eqeq2d y = A B x = y 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 x = A B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌
68 4 67 ceqsexv y y = A B x = y 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 x = A B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌
69 15 65 68 3bitri A B V × V ran V E E 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 V 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 𝖨𝗆𝗀 pprod I 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 x x = A B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌
70 14 3 brco x 𝖡𝗂𝗀𝖼𝗎𝗉 𝖡𝗂𝗀𝖼𝗎𝗉 C y x 𝖡𝗂𝗀𝖼𝗎𝗉 y y 𝖡𝗂𝗀𝖼𝗎𝗉 C
71 16 brbigcup x 𝖡𝗂𝗀𝖼𝗎𝗉 y x = y
72 eqcom x = y y = x
73 71 72 bitri x 𝖡𝗂𝗀𝖼𝗎𝗉 y y = x
74 3 brbigcup y 𝖡𝗂𝗀𝖼𝗎𝗉 C y = C
75 eqcom y = C C = y
76 74 75 bitri y 𝖡𝗂𝗀𝖼𝗎𝗉 C C = y
77 73 76 anbi12i x 𝖡𝗂𝗀𝖼𝗎𝗉 y y 𝖡𝗂𝗀𝖼𝗎𝗉 C y = x C = y
78 77 exbii y x 𝖡𝗂𝗀𝖼𝗎𝗉 y y 𝖡𝗂𝗀𝖼𝗎𝗉 C y y = x C = y
79 vuniex x V
80 unieq y = x y = x
81 80 eqeq2d y = x C = y C = x
82 79 81 ceqsexv y y = x C = y C = x
83 70 78 82 3bitri x 𝖡𝗂𝗀𝖼𝗎𝗉 𝖡𝗂𝗀𝖼𝗎𝗉 C C = x
84 69 83 anbi12i A B V × V ran V E E 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 V 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 𝖨𝗆𝗀 pprod I 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 x x 𝖡𝗂𝗀𝖼𝗎𝗉 𝖡𝗂𝗀𝖼𝗎𝗉 C x = A B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 C = x
85 84 exbii x A B V × V ran V E E 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 V 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 𝖨𝗆𝗀 pprod I 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇 x x 𝖡𝗂𝗀𝖼𝗎𝗉 𝖡𝗂𝗀𝖼𝗎𝗉 C x x = A B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 C = x
86 11 13 85 3bitri A B 𝖠𝗉𝗉𝗅𝗒 C x x = A B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌 C = x
87 dffv5 A B = A B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌
88 87 eqeq2i C = A B C = A B 𝖲𝗂𝗇𝗀𝗅𝖾𝗍𝗈𝗇𝗌
89 9 86 88 3bitr4i A B 𝖠𝗉𝗉𝗅𝗒 C C = A B