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Theorem smfpimbor1lem1 39683
Description: Every open set belongs to 𝑇. This is the second step in the proof of Proposition 121E (f) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
smfpimbor1lem1.s (𝜑𝑆 ∈ SAlg)
smfpimbor1lem1.f (𝜑𝐹 ∈ (SMblFn‘𝑆))
smfpimbor1lem1.a 𝐷 = dom 𝐹
smfpimbor1lem1.j 𝐽 = (topGen‘ran (,))
smfpimbor1lem1.8 (𝜑𝐺𝐽)
smfpimbor1lem1.t 𝑇 = {𝑒 ∈ 𝒫 ℝ ∣ (𝐹𝑒) ∈ (𝑆t 𝐷)}
Assertion
Ref Expression
smfpimbor1lem1 (𝜑𝐺𝑇)
Distinct variable groups:   𝐷,𝑒   𝑒,𝐹   𝑆,𝑒   𝜑,𝑒
Allowed substitution hints:   𝑇(𝑒)   𝐺(𝑒)   𝐽(𝑒)

Proof of Theorem smfpimbor1lem1
Dummy variables 𝑞 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 smfpimbor1lem1.j . . 3 𝐽 = (topGen‘ran (,))
2 smfpimbor1lem1.8 . . 3 (𝜑𝐺𝐽)
31, 2tgqioo2 38621 . 2 (𝜑 → ∃𝑞(𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐺 = 𝑞))
4 simprr 792 . . . . 5 ((𝜑 ∧ (𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐺 = 𝑞)) → 𝐺 = 𝑞)
5 smfpimbor1lem1.s . . . . . . . . 9 (𝜑𝑆 ∈ SAlg)
6 smfpimbor1lem1.f . . . . . . . . 9 (𝜑𝐹 ∈ (SMblFn‘𝑆))
7 smfpimbor1lem1.a . . . . . . . . 9 𝐷 = dom 𝐹
8 smfpimbor1lem1.t . . . . . . . . 9 𝑇 = {𝑒 ∈ 𝒫 ℝ ∣ (𝐹𝑒) ∈ (𝑆t 𝐷)}
95, 6, 7, 8smfresal 39673 . . . . . . . 8 (𝜑𝑇 ∈ SAlg)
109adantr 480 . . . . . . 7 ((𝜑𝑞 ⊆ ((,) “ (ℚ × ℚ))) → 𝑇 ∈ SAlg)
11 iooex 12069 . . . . . . . . . . . 12 (,) ∈ V
1211imaexi 38410 . . . . . . . . . . 11 ((,) “ (ℚ × ℚ)) ∈ V
1312a1i 11 . . . . . . . . . 10 (𝑞 ⊆ ((,) “ (ℚ × ℚ)) → ((,) “ (ℚ × ℚ)) ∈ V)
14 id 22 . . . . . . . . . 10 (𝑞 ⊆ ((,) “ (ℚ × ℚ)) → 𝑞 ⊆ ((,) “ (ℚ × ℚ)))
1513, 14ssexd 4733 . . . . . . . . 9 (𝑞 ⊆ ((,) “ (ℚ × ℚ)) → 𝑞 ∈ V)
1615adantl 481 . . . . . . . 8 ((𝜑𝑞 ⊆ ((,) “ (ℚ × ℚ))) → 𝑞 ∈ V)
17 simpr 476 . . . . . . . . 9 ((𝜑𝑞 ⊆ ((,) “ (ℚ × ℚ))) → 𝑞 ⊆ ((,) “ (ℚ × ℚ)))
18 ioofun 38625 . . . . . . . . . . . . . . 15 Fun (,)
1918a1i 11 . . . . . . . . . . . . . 14 (𝑞 ∈ ((,) “ (ℚ × ℚ)) → Fun (,))
20 id 22 . . . . . . . . . . . . . 14 (𝑞 ∈ ((,) “ (ℚ × ℚ)) → 𝑞 ∈ ((,) “ (ℚ × ℚ)))
21 fvelima 6158 . . . . . . . . . . . . . 14 ((Fun (,) ∧ 𝑞 ∈ ((,) “ (ℚ × ℚ))) → ∃𝑝 ∈ (ℚ × ℚ)((,)‘𝑝) = 𝑞)
2219, 20, 21syl2anc 691 . . . . . . . . . . . . 13 (𝑞 ∈ ((,) “ (ℚ × ℚ)) → ∃𝑝 ∈ (ℚ × ℚ)((,)‘𝑝) = 𝑞)
2322adantl 481 . . . . . . . . . . . 12 ((𝜑𝑞 ∈ ((,) “ (ℚ × ℚ))) → ∃𝑝 ∈ (ℚ × ℚ)((,)‘𝑝) = 𝑞)
24 id 22 . . . . . . . . . . . . . . . . . . . 20 (((,)‘𝑝) = 𝑞 → ((,)‘𝑝) = 𝑞)
2524eqcomd 2616 . . . . . . . . . . . . . . . . . . 19 (((,)‘𝑝) = 𝑞𝑞 = ((,)‘𝑝))
2625adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝑝 ∈ (ℚ × ℚ) ∧ ((,)‘𝑝) = 𝑞) → 𝑞 = ((,)‘𝑝))
27 1st2nd2 7096 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 ∈ (ℚ × ℚ) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
2827fveq2d 6107 . . . . . . . . . . . . . . . . . . . 20 (𝑝 ∈ (ℚ × ℚ) → ((,)‘𝑝) = ((,)‘⟨(1st𝑝), (2nd𝑝)⟩))
29 df-ov 6552 . . . . . . . . . . . . . . . . . . . . . 22 ((1st𝑝)(,)(2nd𝑝)) = ((,)‘⟨(1st𝑝), (2nd𝑝)⟩)
3029eqcomi 2619 . . . . . . . . . . . . . . . . . . . . 21 ((,)‘⟨(1st𝑝), (2nd𝑝)⟩) = ((1st𝑝)(,)(2nd𝑝))
3130a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑝 ∈ (ℚ × ℚ) → ((,)‘⟨(1st𝑝), (2nd𝑝)⟩) = ((1st𝑝)(,)(2nd𝑝)))
3228, 31eqtrd 2644 . . . . . . . . . . . . . . . . . . 19 (𝑝 ∈ (ℚ × ℚ) → ((,)‘𝑝) = ((1st𝑝)(,)(2nd𝑝)))
3332adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝑝 ∈ (ℚ × ℚ) ∧ ((,)‘𝑝) = 𝑞) → ((,)‘𝑝) = ((1st𝑝)(,)(2nd𝑝)))
3426, 33eqtrd 2644 . . . . . . . . . . . . . . . . 17 ((𝑝 ∈ (ℚ × ℚ) ∧ ((,)‘𝑝) = 𝑞) → 𝑞 = ((1st𝑝)(,)(2nd𝑝)))
35343adant1 1072 . . . . . . . . . . . . . . . 16 ((𝜑𝑝 ∈ (ℚ × ℚ) ∧ ((,)‘𝑝) = 𝑞) → 𝑞 = ((1st𝑝)(,)(2nd𝑝)))
36 ioossre 12106 . . . . . . . . . . . . . . . . . . . . 21 ((1st𝑝)(,)(2nd𝑝)) ⊆ ℝ
37 ovex 6577 . . . . . . . . . . . . . . . . . . . . . 22 ((1st𝑝)(,)(2nd𝑝)) ∈ V
3837elpw 4114 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑝)(,)(2nd𝑝)) ∈ 𝒫 ℝ ↔ ((1st𝑝)(,)(2nd𝑝)) ⊆ ℝ)
3936, 38mpbir 220 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑝)(,)(2nd𝑝)) ∈ 𝒫 ℝ
4039a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 ∈ (ℚ × ℚ)) → ((1st𝑝)(,)(2nd𝑝)) ∈ 𝒫 ℝ)
415adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑝 ∈ (ℚ × ℚ)) → 𝑆 ∈ SAlg)
426adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑝 ∈ (ℚ × ℚ)) → 𝐹 ∈ (SMblFn‘𝑆))
43 xp1st 7089 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 ∈ (ℚ × ℚ) → (1st𝑝) ∈ ℚ)
4443qred 38547 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 ∈ (ℚ × ℚ) → (1st𝑝) ∈ ℝ)
4544rexrd 9968 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 ∈ (ℚ × ℚ) → (1st𝑝) ∈ ℝ*)
4645adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑝 ∈ (ℚ × ℚ)) → (1st𝑝) ∈ ℝ*)
47 xp2nd 7090 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 ∈ (ℚ × ℚ) → (2nd𝑝) ∈ ℚ)
4847qred 38547 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 ∈ (ℚ × ℚ) → (2nd𝑝) ∈ ℝ)
4948rexrd 9968 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 ∈ (ℚ × ℚ) → (2nd𝑝) ∈ ℝ*)
5049adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑝 ∈ (ℚ × ℚ)) → (2nd𝑝) ∈ ℝ*)
5141, 42, 7, 46, 50smfpimioo 39672 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 ∈ (ℚ × ℚ)) → (𝐹 “ ((1st𝑝)(,)(2nd𝑝))) ∈ (𝑆t 𝐷))
5240, 51jca 553 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑝 ∈ (ℚ × ℚ)) → (((1st𝑝)(,)(2nd𝑝)) ∈ 𝒫 ℝ ∧ (𝐹 “ ((1st𝑝)(,)(2nd𝑝))) ∈ (𝑆t 𝐷)))
53 imaeq2 5381 . . . . . . . . . . . . . . . . . . . 20 (𝑒 = ((1st𝑝)(,)(2nd𝑝)) → (𝐹𝑒) = (𝐹 “ ((1st𝑝)(,)(2nd𝑝))))
5453eleq1d 2672 . . . . . . . . . . . . . . . . . . 19 (𝑒 = ((1st𝑝)(,)(2nd𝑝)) → ((𝐹𝑒) ∈ (𝑆t 𝐷) ↔ (𝐹 “ ((1st𝑝)(,)(2nd𝑝))) ∈ (𝑆t 𝐷)))
5554, 8elrab2 3333 . . . . . . . . . . . . . . . . . 18 (((1st𝑝)(,)(2nd𝑝)) ∈ 𝑇 ↔ (((1st𝑝)(,)(2nd𝑝)) ∈ 𝒫 ℝ ∧ (𝐹 “ ((1st𝑝)(,)(2nd𝑝))) ∈ (𝑆t 𝐷)))
5652, 55sylibr 223 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (ℚ × ℚ)) → ((1st𝑝)(,)(2nd𝑝)) ∈ 𝑇)
57563adant3 1074 . . . . . . . . . . . . . . . 16 ((𝜑𝑝 ∈ (ℚ × ℚ) ∧ ((,)‘𝑝) = 𝑞) → ((1st𝑝)(,)(2nd𝑝)) ∈ 𝑇)
5835, 57eqeltrd 2688 . . . . . . . . . . . . . . 15 ((𝜑𝑝 ∈ (ℚ × ℚ) ∧ ((,)‘𝑝) = 𝑞) → 𝑞𝑇)
59583exp 1256 . . . . . . . . . . . . . 14 (𝜑 → (𝑝 ∈ (ℚ × ℚ) → (((,)‘𝑝) = 𝑞𝑞𝑇)))
6059rexlimdv 3012 . . . . . . . . . . . . 13 (𝜑 → (∃𝑝 ∈ (ℚ × ℚ)((,)‘𝑝) = 𝑞𝑞𝑇))
6160adantr 480 . . . . . . . . . . . 12 ((𝜑𝑞 ∈ ((,) “ (ℚ × ℚ))) → (∃𝑝 ∈ (ℚ × ℚ)((,)‘𝑝) = 𝑞𝑞𝑇))
6223, 61mpd 15 . . . . . . . . . . 11 ((𝜑𝑞 ∈ ((,) “ (ℚ × ℚ))) → 𝑞𝑇)
6362ssd 38278 . . . . . . . . . 10 (𝜑 → ((,) “ (ℚ × ℚ)) ⊆ 𝑇)
6463adantr 480 . . . . . . . . 9 ((𝜑𝑞 ⊆ ((,) “ (ℚ × ℚ))) → ((,) “ (ℚ × ℚ)) ⊆ 𝑇)
6517, 64sstrd 3578 . . . . . . . 8 ((𝜑𝑞 ⊆ ((,) “ (ℚ × ℚ))) → 𝑞𝑇)
6616, 65elpwd 38264 . . . . . . 7 ((𝜑𝑞 ⊆ ((,) “ (ℚ × ℚ))) → 𝑞 ∈ 𝒫 𝑇)
67 ssdomg 7887 . . . . . . . . . 10 (((,) “ (ℚ × ℚ)) ∈ V → (𝑞 ⊆ ((,) “ (ℚ × ℚ)) → 𝑞 ≼ ((,) “ (ℚ × ℚ))))
6812, 67ax-mp 5 . . . . . . . . 9 (𝑞 ⊆ ((,) “ (ℚ × ℚ)) → 𝑞 ≼ ((,) “ (ℚ × ℚ)))
69 qct 38519 . . . . . . . . . . . . 13 ℚ ≼ ω
7069, 69pm3.2i 470 . . . . . . . . . . . 12 (ℚ ≼ ω ∧ ℚ ≼ ω)
71 xpct 8722 . . . . . . . . . . . 12 ((ℚ ≼ ω ∧ ℚ ≼ ω) → (ℚ × ℚ) ≼ ω)
7270, 71ax-mp 5 . . . . . . . . . . 11 (ℚ × ℚ) ≼ ω
73 fimact 9238 . . . . . . . . . . 11 (((ℚ × ℚ) ≼ ω ∧ Fun (,)) → ((,) “ (ℚ × ℚ)) ≼ ω)
7472, 18, 73mp2an 704 . . . . . . . . . 10 ((,) “ (ℚ × ℚ)) ≼ ω
7574a1i 11 . . . . . . . . 9 (𝑞 ⊆ ((,) “ (ℚ × ℚ)) → ((,) “ (ℚ × ℚ)) ≼ ω)
76 domtr 7895 . . . . . . . . 9 ((𝑞 ≼ ((,) “ (ℚ × ℚ)) ∧ ((,) “ (ℚ × ℚ)) ≼ ω) → 𝑞 ≼ ω)
7768, 75, 76syl2anc 691 . . . . . . . 8 (𝑞 ⊆ ((,) “ (ℚ × ℚ)) → 𝑞 ≼ ω)
7877adantl 481 . . . . . . 7 ((𝜑𝑞 ⊆ ((,) “ (ℚ × ℚ))) → 𝑞 ≼ ω)
7910, 66, 78salunicl 39212 . . . . . 6 ((𝜑𝑞 ⊆ ((,) “ (ℚ × ℚ))) → 𝑞𝑇)
8079adantrr 749 . . . . 5 ((𝜑 ∧ (𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐺 = 𝑞)) → 𝑞𝑇)
814, 80eqeltrd 2688 . . . 4 ((𝜑 ∧ (𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐺 = 𝑞)) → 𝐺𝑇)
8281ex 449 . . 3 (𝜑 → ((𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐺 = 𝑞) → 𝐺𝑇))
8382exlimdv 1848 . 2 (𝜑 → (∃𝑞(𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐺 = 𝑞) → 𝐺𝑇))
843, 83mpd 15 1 (𝜑𝐺𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wex 1695  wcel 1977  wrex 2897  {crab 2900  Vcvv 3173  wss 3540  𝒫 cpw 4108  cop 4131   cuni 4372   class class class wbr 4583   × cxp 5036  ccnv 5037  dom cdm 5038  ran crn 5039  cima 5041  Fun wfun 5798  cfv 5804  (class class class)co 6549  ωcom 6957  1st c1st 7057  2nd c2nd 7058  cdom 7839  cr 9814  *cxr 9952  cq 11664  (,)cioo 12046  t crest 15904  topGenctg 15921  SAlgcsalg 39204  SMblFncsmblfn 39586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cc 9140  ax-ac2 9168  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-acn 8651  df-ac 8822  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-ioo 12050  df-ico 12052  df-fl 12455  df-rest 15906  df-topgen 15927  df-bases 20522  df-salg 39205  df-smblfn 39587
This theorem is referenced by:  smfpimbor1lem2  39684
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