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Theorem pimgtpnf2f 45998
Description: Given a real-valued function, the preimage of an open interval, unbounded above, with lower bound +∞, is the empty set. (Contributed by Glauco Siliprandi, 15-Dec-2021.)
Hypotheses
Ref Expression
pimgtpnf2f.1 β„²π‘₯𝐹
pimgtpnf2f.2 β„²π‘₯𝐴
pimgtpnf2f.3 (πœ‘ β†’ 𝐹:π΄βŸΆβ„)
Assertion
Ref Expression
pimgtpnf2f (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ +∞ < (πΉβ€˜π‘₯)} = βˆ…)

Proof of Theorem pimgtpnf2f
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 pimgtpnf2f.2 . . 3 β„²π‘₯𝐴
2 nfcv 2897 . . 3 Ⅎ𝑦𝐴
3 nfv 1909 . . 3 Ⅎ𝑦+∞ < (πΉβ€˜π‘₯)
4 nfcv 2897 . . . 4 β„²π‘₯+∞
5 nfcv 2897 . . . 4 β„²π‘₯ <
6 pimgtpnf2f.1 . . . . 5 β„²π‘₯𝐹
7 nfcv 2897 . . . . 5 β„²π‘₯𝑦
86, 7nffv 6895 . . . 4 β„²π‘₯(πΉβ€˜π‘¦)
94, 5, 8nfbr 5188 . . 3 β„²π‘₯+∞ < (πΉβ€˜π‘¦)
10 fveq2 6885 . . . 4 (π‘₯ = 𝑦 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))
1110breq2d 5153 . . 3 (π‘₯ = 𝑦 β†’ (+∞ < (πΉβ€˜π‘₯) ↔ +∞ < (πΉβ€˜π‘¦)))
121, 2, 3, 9, 11cbvrabw 3461 . 2 {π‘₯ ∈ 𝐴 ∣ +∞ < (πΉβ€˜π‘₯)} = {𝑦 ∈ 𝐴 ∣ +∞ < (πΉβ€˜π‘¦)}
13 pimgtpnf2f.3 . . . . . . . 8 (πœ‘ β†’ 𝐹:π΄βŸΆβ„)
1413ffvelcdmda 7080 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ 𝐴) β†’ (πΉβ€˜π‘¦) ∈ ℝ)
1514rexrd 11268 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ 𝐴) β†’ (πΉβ€˜π‘¦) ∈ ℝ*)
1615pnfged 44761 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ 𝐴) β†’ (πΉβ€˜π‘¦) ≀ +∞)
17 pnfxr 11272 . . . . . . 7 +∞ ∈ ℝ*
1817a1i 11 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ 𝐴) β†’ +∞ ∈ ℝ*)
1915, 18xrlenltd 11284 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ 𝐴) β†’ ((πΉβ€˜π‘¦) ≀ +∞ ↔ Β¬ +∞ < (πΉβ€˜π‘¦)))
2016, 19mpbid 231 . . . 4 ((πœ‘ ∧ 𝑦 ∈ 𝐴) β†’ Β¬ +∞ < (πΉβ€˜π‘¦))
2120ralrimiva 3140 . . 3 (πœ‘ β†’ βˆ€π‘¦ ∈ 𝐴 Β¬ +∞ < (πΉβ€˜π‘¦))
22 rabeq0 4379 . . 3 ({𝑦 ∈ 𝐴 ∣ +∞ < (πΉβ€˜π‘¦)} = βˆ… ↔ βˆ€π‘¦ ∈ 𝐴 Β¬ +∞ < (πΉβ€˜π‘¦))
2321, 22sylibr 233 . 2 (πœ‘ β†’ {𝑦 ∈ 𝐴 ∣ +∞ < (πΉβ€˜π‘¦)} = βˆ…)
2412, 23eqtrid 2778 1 (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ +∞ < (πΉβ€˜π‘₯)} = βˆ…)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  β„²wnfc 2877  βˆ€wral 3055  {crab 3426  βˆ…c0 4317   class class class wbr 5141  βŸΆwf 6533  β€˜cfv 6537  β„cr 11111  +∞cpnf 11249  β„*cxr 11251   < clt 11252   ≀ cle 11253
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258
This theorem is referenced by:  pimgtpnf2  45999  smfpimgtxr  46073
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