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Theorem suprleubrnmpt 45337
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 2740 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
3 suprleubrnmpt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
41, 2, 3rnmptssd 45103 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ)
5 suprleubrnmpt.a . . . 4 (𝜑𝐴 ≠ ∅)
61, 3, 2, 5rnmptn0 6275 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ≠ ∅)
7 suprleubrnmpt.e . . . 4 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)
81, 7rnmptbdd 45154 . . 3 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑤 ∈ ran (𝑥𝐴𝐵)𝑤𝑦)
9 suprleubrnmpt.c . . 3 (𝜑𝐶 ∈ ℝ)
10 suprleub 12261 . . 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 1373 . 2 (𝜑 → (sup(ran (𝑥𝐴𝐵), ℝ, < ) ≤ 𝐶 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶))
12 nfmpt1 5274 . . . . . . . 8 𝑥(𝑥𝐴𝐵)
1312nfrn 5977 . . . . . . 7 𝑥ran (𝑥𝐴𝐵)
14 nfv 1913 . . . . . . 7 𝑥 𝑧𝐶
1513, 14nfralw 3317 . . . . . 6 𝑥𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶
161, 15nfan 1898 . . . . 5 𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶)
17 simpr 484 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
182elrnmpt1 5983 . . . . . . . . 9 ((𝑥𝐴𝐵 ∈ ℝ) → 𝐵 ∈ ran (𝑥𝐴𝐵))
1917, 3, 18syl2anc 583 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
2019adantlr 714 . . . . . . 7 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶) ∧ 𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
21 simplr 768 . . . . . . 7 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶) ∧ 𝑥𝐴) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶)
22 breq1 5169 . . . . . . . 8 (𝑧 = 𝐵 → (𝑧𝐶𝐵𝐶))
2322rspcva 3633 . . . . . . 7 ((𝐵 ∈ ran (𝑥𝐴𝐵) ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶) → 𝐵𝐶)
2420, 21, 23syl2anc 583 . . . . . 6 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶) ∧ 𝑥𝐴) → 𝐵𝐶)
2524ex 412 . . . . 5 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶) → (𝑥𝐴𝐵𝐶))
2616, 25ralrimi 3263 . . . 4 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶) → ∀𝑥𝐴 𝐵𝐶)
2726ex 412 . . 3 (𝜑 → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶 → ∀𝑥𝐴 𝐵𝐶))
28 vex 3492 . . . . . . . . 9 𝑧 ∈ V
292elrnmpt 5981 . . . . . . . . 9 (𝑧 ∈ V → (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵))
3028, 29ax-mp 5 . . . . . . . 8 (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵)
3130biimpi 216 . . . . . . 7 (𝑧 ∈ ran (𝑥𝐴𝐵) → ∃𝑥𝐴 𝑧 = 𝐵)
3231adantl 481 . . . . . 6 ((∀𝑥𝐴 𝐵𝐶𝑧 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 𝑧 = 𝐵)
33 nfra1 3290 . . . . . . . 8 𝑥𝑥𝐴 𝐵𝐶
34 rspa 3254 . . . . . . . . . 10 ((∀𝑥𝐴 𝐵𝐶𝑥𝐴) → 𝐵𝐶)
3522biimprcd 250 . . . . . . . . . 10 (𝐵𝐶 → (𝑧 = 𝐵𝑧𝐶))
3634, 35syl 17 . . . . . . . . 9 ((∀𝑥𝐴 𝐵𝐶𝑥𝐴) → (𝑧 = 𝐵𝑧𝐶))
3736ex 412 . . . . . . . 8 (∀𝑥𝐴 𝐵𝐶 → (𝑥𝐴 → (𝑧 = 𝐵𝑧𝐶)))
3833, 14, 37rexlimd 3272 . . . . . . 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 3152 . . . 4 (∀𝑥𝐴 𝐵𝐶 → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶)
4241a1i 11 . . 3 (𝜑 → (∀𝑥𝐴 𝐵𝐶 → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶))
4327, 42impbid 212 . 2 (𝜑 → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶 ↔ ∀𝑥𝐴 𝐵𝐶))
4411, 43bitrd 279 1 (𝜑 → (sup(ran (𝑥𝐴𝐵), ℝ, < ) ≤ 𝐶 ↔ ∀𝑥𝐴 𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wnf 1781  wcel 2108  wne 2946  wral 3067  wrex 3076  Vcvv 3488  wss 3976  c0 4352   class class class wbr 5166  cmpt 5249  ran crn 5701  supcsup 9509  cr 11183   < clt 11324  cle 11325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-po 5607  df-so 5608  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-sup 9511  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523
This theorem is referenced by:  smfsuplem1  46732
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