Metamath Proof Explorer


Theorem pimxrneun

Description: The preimage of a set of extended reals that does not contain a value C is the union of the preimage of the elements smaller than C and the preimage of the subset of elements larger than C . (Contributed by Glauco Siliprandi, 21-Dec-2024)

Ref Expression
Hypotheses pimxrneun.1 x φ
pimxrneun.2 φ x A B *
pimxrneun.3 φ x A C *
Assertion pimxrneun φ x A | B C = x A | B < C x A | C < B

Proof

Step Hyp Ref Expression
1 pimxrneun.1 x φ
2 pimxrneun.2 φ x A B *
3 pimxrneun.3 φ x A C *
4 nfrab1 _ x x A | B < C
5 nfrab1 _ x x A | C < B
6 4 5 nfun _ x x A | B < C x A | C < B
7 simpl x A B < C x A
8 simpr x A B < C B < C
9 7 8 jca x A B < C x A B < C
10 rabid x x A | B < C x A B < C
11 9 10 sylibr x A B < C x x A | B < C
12 11 adantll φ x A B < C x x A | B < C
13 elun1 x x A | B < C x x A | B < C x A | C < B
14 12 13 syl φ x A B < C x x A | B < C x A | C < B
15 14 3adantl3 φ x A B C B < C x x A | B < C x A | C < B
16 3simpa φ x A B C φ x A
17 16 adantr φ x A B C ¬ B < C φ x A
18 3 adantr φ x A ¬ B < C C *
19 18 3adantl3 φ x A B C ¬ B < C C *
20 2 adantr φ x A ¬ B < C B *
21 20 3adantl3 φ x A B C ¬ B < C B *
22 simpr φ x A B C ¬ B < C ¬ B < C
23 19 21 22 xrnltled φ x A B C ¬ B < C C B
24 necom B C C B
25 24 biimpi B C C B
26 25 adantr B C ¬ B < C C B
27 26 3ad2antl3 φ x A B C ¬ B < C C B
28 19 21 23 27 xrleneltd φ x A B C ¬ B < C C < B
29 id x A C < B x A C < B
30 29 adantll φ x A C < B x A C < B
31 rabid x x A | C < B x A C < B
32 30 31 sylibr φ x A C < B x x A | C < B
33 elun2 x x A | C < B x x A | B < C x A | C < B
34 32 33 syl φ x A C < B x x A | B < C x A | C < B
35 17 28 34 syl2anc φ x A B C ¬ B < C x x A | B < C x A | C < B
36 15 35 pm2.61dan φ x A B C x x A | B < C x A | C < B
37 1 6 36 rabssd φ x A | B C x A | B < C x A | C < B
38 2 adantr φ x A B < C B *
39 3 adantr φ x A B < C C *
40 simpr φ x A B < C B < C
41 38 39 40 xrltned φ x A B < C B C
42 41 ex φ x A B < C B C
43 1 42 ss2rabdf φ x A | B < C x A | B C
44 3 adantr φ x A C < B C *
45 2 adantr φ x A C < B B *
46 simpr φ x A C < B C < B
47 44 45 46 xrgtned φ x A C < B B C
48 47 ex φ x A C < B B C
49 1 48 ss2rabdf φ x A | C < B x A | B C
50 43 49 unssd φ x A | B < C x A | C < B x A | B C
51 37 50 eqssd φ x A | B C = x A | B < C x A | C < B