Theorem pimgtmnf 40932

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

Hypotheses

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

Assertion

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

Distinct variable group:   x, A

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

Proof of Theorem pimgtmnf

Step	Hyp	Ref	Expression
1		pimgtmnf.1	. . 3 |- F/ x ph
2		eqidd 2623	. . . . . 6 |- ( ph -> ( x e. A |-> B ) = ( x e. A |-> B ) )
3		pimgtmnf.2	. . . . . 6 |- ( ( ph /\ x e. A ) -> B e. RR )
4	2, 3	fvmpt2d 6293	. . . . 5 |- ( ( ph /\ x e. A ) -> ( ( x e. A |-> B ) ` x ) = B )
5	4	eqcomd 2628	. . . 4 |- ( ( ph /\ x e. A ) -> B = ( ( x e. A |-> B ) ` x ) )
6	5	breq2d 4665	. . 3 |- ( ( ph /\ x e. A ) -> ( -oo < B <-> -oo < ( ( x e. A |-> B ) ` x ) ) )
7	1, 6	rabbida 39274	. 2 |- ( ph -> { x e. A | -oo < B } = { x e. A | -oo < ( ( x e. A |-> B ) ` x ) } )
8		nfmpt1 4747	. . 3 |- F/_ x ( x e. A |-> B )
9		eqid 2622	. . . 4 |- ( x e. A |-> B ) = ( x e. A |-> B )
10	1, 3, 9	fmptdf 6387	. . 3 |- ( ph -> ( x e. A |-> B ) : A --> RR )
11	8, 10	pimgtmnf2 40924	. 2 |- ( ph -> { x e. A | -oo < ( ( x e. A |-> B ) ` x ) } = A )
12	7, 11	eqtrd 2656	1 |- ( ph -> { x e. A | -oo < B } = A )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   {crab 2916   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888   RRcr 9935   -oocmnf 10072   < 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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079

This theorem is referenced by:  smfpimgtxr  40988