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Theorem pimgtmnf2 46165
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 4069 . . 3 {π‘₯ ∈ 𝐴 ∣ -∞ < (πΉβ€˜π‘₯)} βŠ† 𝐴
21a1i 11 . 2 (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ -∞ < (πΉβ€˜π‘₯)} βŠ† 𝐴)
3 ssid 3995 . . . . 5 𝐴 βŠ† 𝐴
43a1i 11 . . . 4 (πœ‘ β†’ 𝐴 βŠ† 𝐴)
5 pimgtmnf2.2 . . . . . . . 8 (πœ‘ β†’ 𝐹:π΄βŸΆβ„)
65ffvelcdmda 7089 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ 𝐴) β†’ (πΉβ€˜π‘¦) ∈ ℝ)
76mnfltd 13136 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ 𝐴) β†’ -∞ < (πΉβ€˜π‘¦))
87ralrimiva 3136 . . . . 5 (πœ‘ β†’ βˆ€π‘¦ ∈ 𝐴 -∞ < (πΉβ€˜π‘¦))
9 nfcv 2892 . . . . . . 7 β„²π‘₯-∞
10 nfcv 2892 . . . . . . 7 β„²π‘₯ <
11 pimgtmnf2.1 . . . . . . . 8 β„²π‘₯𝐹
12 nfcv 2892 . . . . . . . 8 β„²π‘₯𝑦
1311, 12nffv 6902 . . . . . . 7 β„²π‘₯(πΉβ€˜π‘¦)
149, 10, 13nfbr 5190 . . . . . 6 β„²π‘₯-∞ < (πΉβ€˜π‘¦)
15 nfv 1909 . . . . . 6 Ⅎ𝑦-∞ < (πΉβ€˜π‘₯)
16 fveq2 6892 . . . . . . 7 (𝑦 = π‘₯ β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π‘₯))
1716breq2d 5155 . . . . . 6 (𝑦 = π‘₯ β†’ (-∞ < (πΉβ€˜π‘¦) ↔ -∞ < (πΉβ€˜π‘₯)))
1814, 15, 17cbvralw 3294 . . . . 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 510 . . 3 (πœ‘ β†’ (𝐴 βŠ† 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 -∞ < (πΉβ€˜π‘₯)))
21 nfcv 2892 . . . 4 β„²π‘₯𝐴
2221, 21ssrabf 44545 . . 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 3990 1 (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ -∞ < (πΉβ€˜π‘₯)} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  β„²wnfc 2875  βˆ€wral 3051  {crab 3419   βŠ† wss 3939   class class class wbr 5143  βŸΆwf 6539  β€˜cfv 6543  β„cr 11137  -∞cmnf 11276   < clt 11278
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 7738  ax-cnex 11194
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 3942  df-un 3944  df-ss 3956  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 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283
This theorem is referenced by: (None)
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