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Theorem suprleubrnmpt 39962
Description: The supremum of a nonempty bounded indexed set of reals is less than or equal to an upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
suprleubrnmpt.x 𝑥𝜑
suprleubrnmpt.a (𝜑𝐴 ≠ ∅)
suprleubrnmpt.b ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
suprleubrnmpt.e (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)
suprleubrnmpt.c (𝜑𝐶 ∈ ℝ)
Assertion
Ref Expression
suprleubrnmpt (𝜑 → (sup(ran (𝑥𝐴𝐵), ℝ, < ) ≤ 𝐶 ↔ ∀𝑥𝐴 𝐵𝐶))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem suprleubrnmpt
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 suprleubrnmpt.x . . . 4 𝑥𝜑
2 eqid 2651 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
3 suprleubrnmpt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
41, 2, 3rnmptssd 39699 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ)
5 suprleubrnmpt.a . . . 4 (𝜑𝐴 ≠ ∅)
61, 3, 2, 5rnmptn0 39727 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ≠ ∅)
7 suprleubrnmpt.e . . . 4 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)
81, 7rnmptbdd 39770 . . 3 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑤 ∈ ran (𝑥𝐴𝐵)𝑤𝑦)
9 suprleubrnmpt.c . . 3 (𝜑𝐶 ∈ ℝ)
10 suprleub 11027 . . 3 (((ran (𝑥𝐴𝐵) ⊆ ℝ ∧ ran (𝑥𝐴𝐵) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑤 ∈ ran (𝑥𝐴𝐵)𝑤𝑦) ∧ 𝐶 ∈ ℝ) → (sup(ran (𝑥𝐴𝐵), ℝ, < ) ≤ 𝐶 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶))
114, 6, 8, 9, 10syl31anc 1369 . 2 (𝜑 → (sup(ran (𝑥𝐴𝐵), ℝ, < ) ≤ 𝐶 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶))
12 nfmpt1 4780 . . . . . . . 8 𝑥(𝑥𝐴𝐵)
1312nfrn 5400 . . . . . . 7 𝑥ran (𝑥𝐴𝐵)
14 nfv 1883 . . . . . . 7 𝑥 𝑧𝐶
1513, 14nfral 2974 . . . . . 6 𝑥𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶
161, 15nfan 1868 . . . . 5 𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶)
17 simpr 476 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
182elrnmpt1 5406 . . . . . . . . 9 ((𝑥𝐴𝐵 ∈ ℝ) → 𝐵 ∈ ran (𝑥𝐴𝐵))
1917, 3, 18syl2anc 694 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
2019adantlr 751 . . . . . . 7 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶) ∧ 𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
21 simplr 807 . . . . . . 7 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶) ∧ 𝑥𝐴) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶)
22 breq1 4688 . . . . . . . 8 (𝑧 = 𝐵 → (𝑧𝐶𝐵𝐶))
2322rspcva 3338 . . . . . . 7 ((𝐵 ∈ ran (𝑥𝐴𝐵) ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶) → 𝐵𝐶)
2420, 21, 23syl2anc 694 . . . . . 6 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶) ∧ 𝑥𝐴) → 𝐵𝐶)
2524ex 449 . . . . 5 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶) → (𝑥𝐴𝐵𝐶))
2616, 25ralrimi 2986 . . . 4 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶) → ∀𝑥𝐴 𝐵𝐶)
2726ex 449 . . 3 (𝜑 → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶 → ∀𝑥𝐴 𝐵𝐶))
28 vex 3234 . . . . . . . . 9 𝑧 ∈ V
292elrnmpt 5404 . . . . . . . . 9 (𝑧 ∈ V → (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵))
3028, 29ax-mp 5 . . . . . . . 8 (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵)
3130biimpi 206 . . . . . . 7 (𝑧 ∈ ran (𝑥𝐴𝐵) → ∃𝑥𝐴 𝑧 = 𝐵)
3231adantl 481 . . . . . 6 ((∀𝑥𝐴 𝐵𝐶𝑧 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 𝑧 = 𝐵)
33 nfra1 2970 . . . . . . . 8 𝑥𝑥𝐴 𝐵𝐶
34 rspa 2959 . . . . . . . . . 10 ((∀𝑥𝐴 𝐵𝐶𝑥𝐴) → 𝐵𝐶)
3522biimprcd 240 . . . . . . . . . 10 (𝐵𝐶 → (𝑧 = 𝐵𝑧𝐶))
3634, 35syl 17 . . . . . . . . 9 ((∀𝑥𝐴 𝐵𝐶𝑥𝐴) → (𝑧 = 𝐵𝑧𝐶))
3736ex 449 . . . . . . . 8 (∀𝑥𝐴 𝐵𝐶 → (𝑥𝐴 → (𝑧 = 𝐵𝑧𝐶)))
3833, 14, 37rexlimd 3055 . . . . . . 7 (∀𝑥𝐴 𝐵𝐶 → (∃𝑥𝐴 𝑧 = 𝐵𝑧𝐶))
3938adantr 480 . . . . . 6 ((∀𝑥𝐴 𝐵𝐶𝑧 ∈ ran (𝑥𝐴𝐵)) → (∃𝑥𝐴 𝑧 = 𝐵𝑧𝐶))
4032, 39mpd 15 . . . . 5 ((∀𝑥𝐴 𝐵𝐶𝑧 ∈ ran (𝑥𝐴𝐵)) → 𝑧𝐶)
4140ralrimiva 2995 . . . 4 (∀𝑥𝐴 𝐵𝐶 → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶)
4241a1i 11 . . 3 (𝜑 → (∀𝑥𝐴 𝐵𝐶 → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶))
4327, 42impbid 202 . 2 (𝜑 → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶 ↔ ∀𝑥𝐴 𝐵𝐶))
4411, 43bitrd 268 1 (𝜑 → (sup(ran (𝑥𝐴𝐵), ℝ, < ) ≤ 𝐶 ↔ ∀𝑥𝐴 𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wnf 1748  wcel 2030  wne 2823  wral 2941  wrex 2942  Vcvv 3231  wss 3607  c0 3948   class class class wbr 4685  cmpt 4762  ran crn 5144  supcsup 8387  cr 9973   < clt 10112  cle 10113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307
This theorem is referenced by:  smfsuplem1  41338
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