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Theorem pimltpnf2f 46162
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 2892 . . 3 Ⅎ𝑦𝐴
3 nfv 1909 . . 3 Ⅎ𝑦(πΉβ€˜π‘₯) < +∞
4 pimltpnf2f.1 . . . . 5 β„²π‘₯𝐹
5 nfcv 2892 . . . . 5 β„²π‘₯𝑦
64, 5nffv 6901 . . . 4 β„²π‘₯(πΉβ€˜π‘¦)
7 nfcv 2892 . . . 4 β„²π‘₯ <
8 nfcv 2892 . . . 4 β„²π‘₯+∞
96, 7, 8nfbr 5190 . . 3 β„²π‘₯(πΉβ€˜π‘¦) < +∞
10 fveq2 6891 . . . 4 (π‘₯ = 𝑦 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))
1110breq1d 5153 . . 3 (π‘₯ = 𝑦 β†’ ((πΉβ€˜π‘₯) < +∞ ↔ (πΉβ€˜π‘¦) < +∞))
121, 2, 3, 9, 11cbvrabw 3456 . 2 {π‘₯ ∈ 𝐴 ∣ (πΉβ€˜π‘₯) < +∞} = {𝑦 ∈ 𝐴 ∣ (πΉβ€˜π‘¦) < +∞}
13 nfv 1909 . . 3 β„²π‘¦πœ‘
14 pimltpnf2f.3 . . . 4 (πœ‘ β†’ 𝐹:π΄βŸΆβ„)
1514ffvelcdmda 7088 . . 3 ((πœ‘ ∧ 𝑦 ∈ 𝐴) β†’ (πΉβ€˜π‘¦) ∈ ℝ)
1613, 15pimltpnf 46154 . 2 (πœ‘ β†’ {𝑦 ∈ 𝐴 ∣ (πΉβ€˜π‘¦) < +∞} = 𝐴)
1712, 16eqtrid 2777 1 (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ (πΉβ€˜π‘₯) < +∞} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533  β„²wnfc 2875  {crab 3419   class class class wbr 5143  βŸΆwf 6538  β€˜cfv 6542  β„cr 11135  +∞cpnf 11273   < clt 11276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-cnex 11192
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-pnf 11278  df-xr 11280  df-ltxr 11281
This theorem is referenced by:  pimltpnf2  46163  smfpimltxr  46197
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