Metamath Proof Explorer


Theorem mptprop

Description: Rewrite pairs of ordered pairs as mapping to functions. (Contributed by Thierry Arnoux, 24-Sep-2023)

Ref Expression
Hypotheses brprop.a φ A V
brprop.b φ B W
brprop.c φ C V
brprop.d φ D W
mptprop.1 φ A C
Assertion mptprop φ A B C D = x A C if x = A B D

Proof

Step Hyp Ref Expression
1 brprop.a φ A V
2 brprop.b φ B W
3 brprop.c φ C V
4 brprop.d φ D W
5 mptprop.1 φ A C
6 df-pr A B C D = A B C D
7 fmptsn A V B W A B = x A B
8 1 2 7 syl2anc φ A B = x A B
9 incom A A C = A C A
10 prid1g A V A A C
11 snssi A A C A A C
12 1 10 11 3syl φ A A C
13 df-ss A A C A A C = A
14 12 13 sylib φ A A C = A
15 9 14 eqtr3id φ A C A = A
16 15 mpteq1d φ x A C A B = x A B
17 8 16 eqtr4d φ A B = x A C A B
18 fmptsn C V D W C D = x C D
19 3 4 18 syl2anc φ C D = x C D
20 difprsn1 A C A C A = C
21 5 20 syl φ A C A = C
22 21 mpteq1d φ x A C A D = x C D
23 19 22 eqtr4d φ C D = x A C A D
24 17 23 uneq12d φ A B C D = x A C A B x A C A D
25 partfun x A C if x A B D = x A C A B x A C A D
26 24 25 eqtr4di φ A B C D = x A C if x A B D
27 elsn2g A V x A x = A
28 1 27 syl φ x A x = A
29 28 ifbid φ if x A B D = if x = A B D
30 29 mpteq2dv φ x A C if x A B D = x A C if x = A B D
31 26 30 eqtrd φ A B C D = x A C if x = A B D
32 6 31 syl5eq φ A B C D = x A C if x = A B D