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Theorem pimgtmnf2 45041
Description: Given a real-valued function, the preimage of an open interval, unbounded above, with lower bound -∞, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
pimgtmnf2.1 β„²π‘₯𝐹
pimgtmnf2.2 (πœ‘ β†’ 𝐹:π΄βŸΆβ„)
Assertion
Ref Expression
pimgtmnf2 (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ -∞ < (πΉβ€˜π‘₯)} = 𝐴)
Distinct variable group:   π‘₯,𝐴
Allowed substitution hints:   πœ‘(π‘₯)   𝐹(π‘₯)

Proof of Theorem pimgtmnf2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssrab2 4038 . . 3 {π‘₯ ∈ 𝐴 ∣ -∞ < (πΉβ€˜π‘₯)} βŠ† 𝐴
21a1i 11 . 2 (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ -∞ < (πΉβ€˜π‘₯)} βŠ† 𝐴)
3 ssid 3967 . . . . 5 𝐴 βŠ† 𝐴
43a1i 11 . . . 4 (πœ‘ β†’ 𝐴 βŠ† 𝐴)
5 pimgtmnf2.2 . . . . . . . 8 (πœ‘ β†’ 𝐹:π΄βŸΆβ„)
65ffvelcdmda 7036 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ 𝐴) β†’ (πΉβ€˜π‘¦) ∈ ℝ)
76mnfltd 13050 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ 𝐴) β†’ -∞ < (πΉβ€˜π‘¦))
87ralrimiva 3140 . . . . 5 (πœ‘ β†’ βˆ€π‘¦ ∈ 𝐴 -∞ < (πΉβ€˜π‘¦))
9 nfcv 2904 . . . . . . 7 β„²π‘₯-∞
10 nfcv 2904 . . . . . . 7 β„²π‘₯ <
11 pimgtmnf2.1 . . . . . . . 8 β„²π‘₯𝐹
12 nfcv 2904 . . . . . . . 8 β„²π‘₯𝑦
1311, 12nffv 6853 . . . . . . 7 β„²π‘₯(πΉβ€˜π‘¦)
149, 10, 13nfbr 5153 . . . . . 6 β„²π‘₯-∞ < (πΉβ€˜π‘¦)
15 nfv 1918 . . . . . 6 Ⅎ𝑦-∞ < (πΉβ€˜π‘₯)
16 fveq2 6843 . . . . . . 7 (𝑦 = π‘₯ β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π‘₯))
1716breq2d 5118 . . . . . 6 (𝑦 = π‘₯ β†’ (-∞ < (πΉβ€˜π‘¦) ↔ -∞ < (πΉβ€˜π‘₯)))
1814, 15, 17cbvralw 3288 . . . . 5 (βˆ€π‘¦ ∈ 𝐴 -∞ < (πΉβ€˜π‘¦) ↔ βˆ€π‘₯ ∈ 𝐴 -∞ < (πΉβ€˜π‘₯))
198, 18sylib 217 . . . 4 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 -∞ < (πΉβ€˜π‘₯))
204, 19jca 513 . . 3 (πœ‘ β†’ (𝐴 βŠ† 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 -∞ < (πΉβ€˜π‘₯)))
21 nfcv 2904 . . . 4 β„²π‘₯𝐴
2221, 21ssrabf 43412 . . 3 (𝐴 βŠ† {π‘₯ ∈ 𝐴 ∣ -∞ < (πΉβ€˜π‘₯)} ↔ (𝐴 βŠ† 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 -∞ < (πΉβ€˜π‘₯)))
2320, 22sylibr 233 . 2 (πœ‘ β†’ 𝐴 βŠ† {π‘₯ ∈ 𝐴 ∣ -∞ < (πΉβ€˜π‘₯)})
242, 23eqssd 3962 1 (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ -∞ < (πΉβ€˜π‘₯)} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  β„²wnfc 2884  βˆ€wral 3061  {crab 3406   βŠ† wss 3911   class class class wbr 5106  βŸΆwf 6493  β€˜cfv 6497  β„cr 11055  -∞cmnf 11192   < 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-mnf 11197  df-xr 11198  df-ltxr 11199
This theorem is referenced by: (None)
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