Theorem pimltpnf 40916

Description: Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound +oo, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
pimltpnf.1	|- F/ x ph
pimltpnf.2	|- ( ( ph /\ x e. A ) -> B e. RR )

Assertion

Ref	Expression
pimltpnf	|- ( ph -> { x e. A | B < +oo } = A )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   B( x)

Proof of Theorem pimltpnf

Step	Hyp	Ref	Expression
1		ssrab2 3687	. . 3 |- { x e. A | B < +oo } C_ A
2	1	a1i 11	. 2 |- ( ph -> { x e. A | B < +oo } C_ A )
3		pimltpnf.1	. . . 4 |- F/ x ph
4		simpr 477	. . . . . . 7 |- ( ( ph /\ x e. A ) -> x e. A )
5		pimltpnf.2	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> B e. RR )
6		ltpnf 11954	. . . . . . . 8 |- ( B e. RR -> B < +oo )
7	5, 6	syl 17	. . . . . . 7 |- ( ( ph /\ x e. A ) -> B < +oo )
8	4, 7	jca 554	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( x e. A /\ B < +oo ) )
9		rabid 3116	. . . . . 6 |- ( x e. { x e. A | B < +oo } <-> ( x e. A /\ B < +oo ) )
10	8, 9	sylibr 224	. . . . 5 |- ( ( ph /\ x e. A ) -> x e. { x e. A | B < +oo } )
11	10	ex 450	. . . 4 |- ( ph -> ( x e. A -> x e. { x e. A | B < +oo } ) )
12	3, 11	ralrimi 2957	. . 3 |- ( ph -> A. x e. A x e. { x e. A | B < +oo } )
13		nfcv 2764	. . . 4 |- F/_ x A
14		nfrab1 3122	. . . 4 |- F/_ x { x e. A | B < +oo }
15	13, 14	dfss3f 3595	. . 3 |- ( A C_ { x e. A | B < +oo } <-> A. x e. A x e. { x e. A | B < +oo } )
16	12, 15	sylibr 224	. 2 |- ( ph -> A C_ { x e. A | B < +oo } )
17	2, 16	eqssd 3620	1 |- ( ph -> { x e. A | B < +oo } = A )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   A.wral 2912   {crab 2916   C_ wss 3574   class class class wbr 4653   RRcr 9935   +oocpnf 10071   < clt 10074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-pnf 10076  df-xr 10078  df-ltxr 10079

This theorem is referenced by:  pimltpnf2  40923