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Theorem pimconstlt1 46359
Description: Given a constant function, its preimage with respect to an unbounded below, open interval, with upper bound larger than the constant, is the whole domain. First part of Proposition 121E (a) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
pimconstlt1.1 𝑥𝜑
pimconstlt1.2 (𝜑𝐵 ∈ ℝ)
pimconstlt1.3 𝐹 = (𝑥𝐴𝐵)
pimconstlt1.4 (𝜑𝐵 < 𝐶)
Assertion
Ref Expression
pimconstlt1 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} = 𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐹(𝑥)

Proof of Theorem pimconstlt1
StepHypRef Expression
1 ssrab2 4073 . . 3 {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} ⊆ 𝐴
21a1i 11 . 2 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} ⊆ 𝐴)
3 pimconstlt1.1 . . . 4 𝑥𝜑
4 simpr 483 . . . . . . 7 ((𝜑𝑥𝐴) → 𝑥𝐴)
5 pimconstlt1.3 . . . . . . . . . 10 𝐹 = (𝑥𝐴𝐵)
65a1i 11 . . . . . . . . 9 (𝜑𝐹 = (𝑥𝐴𝐵))
7 pimconstlt1.2 . . . . . . . . . 10 (𝜑𝐵 ∈ ℝ)
87adantr 479 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
96, 8fvmpt2d 7014 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
10 pimconstlt1.4 . . . . . . . . 9 (𝜑𝐵 < 𝐶)
1110adantr 479 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 < 𝐶)
129, 11eqbrtrd 5167 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐹𝑥) < 𝐶)
134, 12jca 510 . . . . . 6 ((𝜑𝑥𝐴) → (𝑥𝐴 ∧ (𝐹𝑥) < 𝐶))
14 rabid 3440 . . . . . 6 (𝑥 ∈ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} ↔ (𝑥𝐴 ∧ (𝐹𝑥) < 𝐶))
1513, 14sylibr 233 . . . . 5 ((𝜑𝑥𝐴) → 𝑥 ∈ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶})
1615ex 411 . . . 4 (𝜑 → (𝑥𝐴𝑥 ∈ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶}))
173, 16ralrimi 3245 . . 3 (𝜑 → ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶})
18 nfcv 2892 . . . 4 𝑥𝐴
19 nfrab1 3439 . . . 4 𝑥{𝑥𝐴 ∣ (𝐹𝑥) < 𝐶}
2018, 19dfss3f 3970 . . 3 (𝐴 ⊆ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} ↔ ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶})
2117, 20sylibr 233 . 2 (𝜑𝐴 ⊆ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶})
222, 21eqssd 3996 1 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wnf 1778  wcel 2099  wral 3051  {crab 3419  wss 3946   class class class wbr 5145  cmpt 5228  cfv 6546  cr 11148   < clt 11289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fv 6554
This theorem is referenced by:  smfconst  46406
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