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Theorem pimgtmnf2 46002
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 4072 . . 3 {π‘₯ ∈ 𝐴 ∣ -∞ < (πΉβ€˜π‘₯)} βŠ† 𝐴
21a1i 11 . 2 (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ -∞ < (πΉβ€˜π‘₯)} βŠ† 𝐴)
3 ssid 3999 . . . . 5 𝐴 βŠ† 𝐴
43a1i 11 . . . 4 (πœ‘ β†’ 𝐴 βŠ† 𝐴)
5 pimgtmnf2.2 . . . . . . . 8 (πœ‘ β†’ 𝐹:π΄βŸΆβ„)
65ffvelcdmda 7080 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ 𝐴) β†’ (πΉβ€˜π‘¦) ∈ ℝ)
76mnfltd 13110 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ 𝐴) β†’ -∞ < (πΉβ€˜π‘¦))
87ralrimiva 3140 . . . . 5 (πœ‘ β†’ βˆ€π‘¦ ∈ 𝐴 -∞ < (πΉβ€˜π‘¦))
9 nfcv 2897 . . . . . . 7 β„²π‘₯-∞
10 nfcv 2897 . . . . . . 7 β„²π‘₯ <
11 pimgtmnf2.1 . . . . . . . 8 β„²π‘₯𝐹
12 nfcv 2897 . . . . . . . 8 β„²π‘₯𝑦
1311, 12nffv 6895 . . . . . . 7 β„²π‘₯(πΉβ€˜π‘¦)
149, 10, 13nfbr 5188 . . . . . 6 β„²π‘₯-∞ < (πΉβ€˜π‘¦)
15 nfv 1909 . . . . . 6 Ⅎ𝑦-∞ < (πΉβ€˜π‘₯)
16 fveq2 6885 . . . . . . 7 (𝑦 = π‘₯ β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π‘₯))
1716breq2d 5153 . . . . . 6 (𝑦 = π‘₯ β†’ (-∞ < (πΉβ€˜π‘¦) ↔ -∞ < (πΉβ€˜π‘₯)))
1814, 15, 17cbvralw 3297 . . . . 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 511 . . 3 (πœ‘ β†’ (𝐴 βŠ† 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 -∞ < (πΉβ€˜π‘₯)))
21 nfcv 2897 . . . 4 β„²π‘₯𝐴
2221, 21ssrabf 44378 . . 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 3994 1 (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ -∞ < (πΉβ€˜π‘₯)} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  β„²wnfc 2877  βˆ€wral 3055  {crab 3426   βŠ† wss 3943   class class class wbr 5141  βŸΆwf 6533  β€˜cfv 6537  β„cr 11111  -∞cmnf 11250   < clt 11252
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
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-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
This theorem is referenced by: (None)
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