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Theorem supxrleubrnmpt 41835
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 2821 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
3 supxrleubrnmpt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
41, 2, 3rnmptssd 41614 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ*)
5 supxrleubrnmpt.c . . 3 (𝜑𝐶 ∈ ℝ*)
6 supxrleub 12697 . . 3 ((ran (𝑥𝐴𝐵) ⊆ ℝ*𝐶 ∈ ℝ*) → (sup(ran (𝑥𝐴𝐵), ℝ*, < ) ≤ 𝐶 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶))
74, 5, 6syl2anc 587 . 2 (𝜑 → (sup(ran (𝑥𝐴𝐵), ℝ*, < ) ≤ 𝐶 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶))
8 nfmpt1 5137 . . . . . . . 8 𝑥(𝑥𝐴𝐵)
98nfrn 5797 . . . . . . 7 𝑥ran (𝑥𝐴𝐵)
10 nfv 1916 . . . . . . 7 𝑥 𝑧𝐶
119, 10nfralw 3213 . . . . . 6 𝑥𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶
121, 11nfan 1901 . . . . 5 𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶)
13 simpr 488 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
142elrnmpt1 5803 . . . . . . . . 9 ((𝑥𝐴𝐵 ∈ ℝ*) → 𝐵 ∈ ran (𝑥𝐴𝐵))
1513, 3, 14syl2anc 587 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
1615adantlr 714 . . . . . . 7 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶) ∧ 𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
17 simplr 768 . . . . . . 7 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶) ∧ 𝑥𝐴) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶)
18 breq1 5042 . . . . . . . 8 (𝑧 = 𝐵 → (𝑧𝐶𝐵𝐶))
1918rspcva 3598 . . . . . . 7 ((𝐵 ∈ ran (𝑥𝐴𝐵) ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶) → 𝐵𝐶)
2016, 17, 19syl2anc 587 . . . . . 6 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶) ∧ 𝑥𝐴) → 𝐵𝐶)
2120ex 416 . . . . 5 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶) → (𝑥𝐴𝐵𝐶))
2212, 21ralrimi 3204 . . . 4 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶) → ∀𝑥𝐴 𝐵𝐶)
2322ex 416 . . 3 (𝜑 → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶 → ∀𝑥𝐴 𝐵𝐶))
24 vex 3474 . . . . . . . . 9 𝑧 ∈ V
252elrnmpt 5801 . . . . . . . . 9 (𝑧 ∈ V → (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵))
2624, 25ax-mp 5 . . . . . . . 8 (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵)
2726biimpi 219 . . . . . . 7 (𝑧 ∈ ran (𝑥𝐴𝐵) → ∃𝑥𝐴 𝑧 = 𝐵)
2827adantl 485 . . . . . 6 ((∀𝑥𝐴 𝐵𝐶𝑧 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 𝑧 = 𝐵)
29 nfra1 3207 . . . . . . . 8 𝑥𝑥𝐴 𝐵𝐶
30 rspa 3194 . . . . . . . . . 10 ((∀𝑥𝐴 𝐵𝐶𝑥𝐴) → 𝐵𝐶)
3118biimprcd 253 . . . . . . . . . 10 (𝐵𝐶 → (𝑧 = 𝐵𝑧𝐶))
3230, 31syl 17 . . . . . . . . 9 ((∀𝑥𝐴 𝐵𝐶𝑥𝐴) → (𝑧 = 𝐵𝑧𝐶))
3332ex 416 . . . . . . . 8 (∀𝑥𝐴 𝐵𝐶 → (𝑥𝐴 → (𝑧 = 𝐵𝑧𝐶)))
3429, 10, 33rexlimd 3303 . . . . . . 7 (∀𝑥𝐴 𝐵𝐶 → (∃𝑥𝐴 𝑧 = 𝐵𝑧𝐶))
3534adantr 484 . . . . . 6 ((∀𝑥𝐴 𝐵𝐶𝑧 ∈ ran (𝑥𝐴𝐵)) → (∃𝑥𝐴 𝑧 = 𝐵𝑧𝐶))
3628, 35mpd 15 . . . . 5 ((∀𝑥𝐴 𝐵𝐶𝑧 ∈ ran (𝑥𝐴𝐵)) → 𝑧𝐶)
3736ralrimiva 3170 . . . 4 (∀𝑥𝐴 𝐵𝐶 → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶)
3837a1i 11 . . 3 (𝜑 → (∀𝑥𝐴 𝐵𝐶 → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶))
3923, 38impbid 215 . 2 (𝜑 → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶 ↔ ∀𝑥𝐴 𝐵𝐶))
407, 39bitrd 282 1 (𝜑 → (sup(ran (𝑥𝐴𝐵), ℝ*, < ) ≤ 𝐶 ↔ ∀𝑥𝐴 𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wnf 1785  wcel 2115  wral 3126  wrex 3127  Vcvv 3471  wss 3910   class class class wbr 5039  cmpt 5119  ran crn 5529  supcsup 8880  *cxr 10651   < clt 10652  cle 10653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591  ax-pre-sup 10592
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-po 5447  df-so 5448  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-er 8264  df-en 8485  df-dom 8486  df-sdom 8487  df-sup 8882  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850
This theorem is referenced by:  supxrleubrnmptf  41883
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