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Theorem pimltpnf2f 45039
Description: Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound +∞, is the whole domain. (Contributed by Glauco Siliprandi, 15-Dec-2024.)
Hypotheses
Ref Expression
pimltpnf2f.1 β„²π‘₯𝐹
pimltpnf2f.2 β„²π‘₯𝐴
pimltpnf2f.3 (πœ‘ β†’ 𝐹:π΄βŸΆβ„)
Assertion
Ref Expression
pimltpnf2f (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ (πΉβ€˜π‘₯) < +∞} = 𝐴)

Proof of Theorem pimltpnf2f
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 pimltpnf2f.2 . . 3 β„²π‘₯𝐴
2 nfcv 2904 . . 3 Ⅎ𝑦𝐴
3 nfv 1918 . . 3 Ⅎ𝑦(πΉβ€˜π‘₯) < +∞
4 pimltpnf2f.1 . . . . 5 β„²π‘₯𝐹
5 nfcv 2904 . . . . 5 β„²π‘₯𝑦
64, 5nffv 6853 . . . 4 β„²π‘₯(πΉβ€˜π‘¦)
7 nfcv 2904 . . . 4 β„²π‘₯ <
8 nfcv 2904 . . . 4 β„²π‘₯+∞
96, 7, 8nfbr 5153 . . 3 β„²π‘₯(πΉβ€˜π‘¦) < +∞
10 fveq2 6843 . . . 4 (π‘₯ = 𝑦 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))
1110breq1d 5116 . . 3 (π‘₯ = 𝑦 β†’ ((πΉβ€˜π‘₯) < +∞ ↔ (πΉβ€˜π‘¦) < +∞))
121, 2, 3, 9, 11cbvrabw 3438 . 2 {π‘₯ ∈ 𝐴 ∣ (πΉβ€˜π‘₯) < +∞} = {𝑦 ∈ 𝐴 ∣ (πΉβ€˜π‘¦) < +∞}
13 nfv 1918 . . 3 β„²π‘¦πœ‘
14 pimltpnf2f.3 . . . 4 (πœ‘ β†’ 𝐹:π΄βŸΆβ„)
1514ffvelcdmda 7036 . . 3 ((πœ‘ ∧ 𝑦 ∈ 𝐴) β†’ (πΉβ€˜π‘¦) ∈ ℝ)
1613, 15pimltpnf 45031 . 2 (πœ‘ β†’ {𝑦 ∈ 𝐴 ∣ (πΉβ€˜π‘¦) < +∞} = 𝐴)
1712, 16eqtrid 2785 1 (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ (πΉβ€˜π‘₯) < +∞} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542  β„²wnfc 2884  {crab 3406   class class class wbr 5106  βŸΆwf 6493  β€˜cfv 6497  β„cr 11055  +∞cpnf 11191   < clt 11194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-pnf 11196  df-xr 11198  df-ltxr 11199
This theorem is referenced by:  pimltpnf2  45040  smfpimltxr  45074
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