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Theorem supxrleubrnmpt 40099
Description: The supremum of a nonempty bounded indexed set of extended reals is less than or equal to an upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
supxrleubrnmpt.x 𝑥𝜑
supxrleubrnmpt.b ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
supxrleubrnmpt.c (𝜑𝐶 ∈ ℝ*)
Assertion
Ref Expression
supxrleubrnmpt (𝜑 → (sup(ran (𝑥𝐴𝐵), ℝ*, < ) ≤ 𝐶 ↔ ∀𝑥𝐴 𝐵𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem supxrleubrnmpt
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 supxrleubrnmpt.x . . . 4 𝑥𝜑
2 eqid 2748 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
3 supxrleubrnmpt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
41, 2, 3rnmptssd 39853 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ*)
5 supxrleubrnmpt.c . . 3 (𝜑𝐶 ∈ ℝ*)
6 supxrleub 12320 . . 3 ((ran (𝑥𝐴𝐵) ⊆ ℝ*𝐶 ∈ ℝ*) → (sup(ran (𝑥𝐴𝐵), ℝ*, < ) ≤ 𝐶 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶))
74, 5, 6syl2anc 696 . 2 (𝜑 → (sup(ran (𝑥𝐴𝐵), ℝ*, < ) ≤ 𝐶 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶))
8 nfmpt1 4887 . . . . . . . 8 𝑥(𝑥𝐴𝐵)
98nfrn 5511 . . . . . . 7 𝑥ran (𝑥𝐴𝐵)
10 nfv 1980 . . . . . . 7 𝑥 𝑧𝐶
119, 10nfral 3071 . . . . . 6 𝑥𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶
121, 11nfan 1965 . . . . 5 𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶)
13 simpr 479 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
142elrnmpt1 5517 . . . . . . . . 9 ((𝑥𝐴𝐵 ∈ ℝ*) → 𝐵 ∈ ran (𝑥𝐴𝐵))
1513, 3, 14syl2anc 696 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
1615adantlr 753 . . . . . . 7 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶) ∧ 𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
17 simplr 809 . . . . . . 7 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶) ∧ 𝑥𝐴) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶)
18 breq1 4795 . . . . . . . 8 (𝑧 = 𝐵 → (𝑧𝐶𝐵𝐶))
1918rspcva 3435 . . . . . . 7 ((𝐵 ∈ ran (𝑥𝐴𝐵) ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶) → 𝐵𝐶)
2016, 17, 19syl2anc 696 . . . . . 6 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶) ∧ 𝑥𝐴) → 𝐵𝐶)
2120ex 449 . . . . 5 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶) → (𝑥𝐴𝐵𝐶))
2212, 21ralrimi 3083 . . . 4 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶) → ∀𝑥𝐴 𝐵𝐶)
2322ex 449 . . 3 (𝜑 → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶 → ∀𝑥𝐴 𝐵𝐶))
24 vex 3331 . . . . . . . . 9 𝑧 ∈ V
252elrnmpt 5515 . . . . . . . . 9 (𝑧 ∈ V → (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵))
2624, 25ax-mp 5 . . . . . . . 8 (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵)
2726biimpi 206 . . . . . . 7 (𝑧 ∈ ran (𝑥𝐴𝐵) → ∃𝑥𝐴 𝑧 = 𝐵)
2827adantl 473 . . . . . 6 ((∀𝑥𝐴 𝐵𝐶𝑧 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 𝑧 = 𝐵)
29 nfra1 3067 . . . . . . . 8 𝑥𝑥𝐴 𝐵𝐶
30 rspa 3056 . . . . . . . . . 10 ((∀𝑥𝐴 𝐵𝐶𝑥𝐴) → 𝐵𝐶)
3118biimprcd 240 . . . . . . . . . 10 (𝐵𝐶 → (𝑧 = 𝐵𝑧𝐶))
3230, 31syl 17 . . . . . . . . 9 ((∀𝑥𝐴 𝐵𝐶𝑥𝐴) → (𝑧 = 𝐵𝑧𝐶))
3332ex 449 . . . . . . . 8 (∀𝑥𝐴 𝐵𝐶 → (𝑥𝐴 → (𝑧 = 𝐵𝑧𝐶)))
3429, 10, 33rexlimd 3152 . . . . . . 7 (∀𝑥𝐴 𝐵𝐶 → (∃𝑥𝐴 𝑧 = 𝐵𝑧𝐶))
3534adantr 472 . . . . . 6 ((∀𝑥𝐴 𝐵𝐶𝑧 ∈ ran (𝑥𝐴𝐵)) → (∃𝑥𝐴 𝑧 = 𝐵𝑧𝐶))
3628, 35mpd 15 . . . . 5 ((∀𝑥𝐴 𝐵𝐶𝑧 ∈ ran (𝑥𝐴𝐵)) → 𝑧𝐶)
3736ralrimiva 3092 . . . 4 (∀𝑥𝐴 𝐵𝐶 → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶)
3837a1i 11 . . 3 (𝜑 → (∀𝑥𝐴 𝐵𝐶 → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶))
3923, 38impbid 202 . 2 (𝜑 → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶 ↔ ∀𝑥𝐴 𝐵𝐶))
407, 39bitrd 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 1620  wnf 1845  wcel 2127  wral 3038  wrex 3039  Vcvv 3328  wss 3703   class class class wbr 4792  cmpt 4869  ran crn 5255  supcsup 8499  *cxr 10236   < clt 10237  cle 10238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102  ax-cnex 10155  ax-resscn 10156  ax-1cn 10157  ax-icn 10158  ax-addcl 10159  ax-addrcl 10160  ax-mulcl 10161  ax-mulrcl 10162  ax-mulcom 10163  ax-addass 10164  ax-mulass 10165  ax-distr 10166  ax-i2m1 10167  ax-1ne0 10168  ax-1rid 10169  ax-rnegex 10170  ax-rrecex 10171  ax-cnre 10172  ax-pre-lttri 10173  ax-pre-lttrn 10174  ax-pre-ltadd 10175  ax-pre-mulgt0 10176  ax-pre-sup 10177
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-nel 3024  df-ral 3043  df-rex 3044  df-reu 3045  df-rmo 3046  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-po 5175  df-so 5176  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-riota 6762  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-er 7899  df-en 8110  df-dom 8111  df-sdom 8112  df-sup 8501  df-pnf 10239  df-mnf 10240  df-xr 10241  df-ltxr 10242  df-le 10243  df-sub 10431  df-neg 10432
This theorem is referenced by:  supxrleubrnmptf  40147
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