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Theorem supxrleubrnmpt 43727
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 2733 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
3 supxrleubrnmpt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
41, 2, 3rnmptssd 43504 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ*)
5 supxrleubrnmpt.c . . 3 (𝜑𝐶 ∈ ℝ*)
6 supxrleub 13251 . . 3 ((ran (𝑥𝐴𝐵) ⊆ ℝ*𝐶 ∈ ℝ*) → (sup(ran (𝑥𝐴𝐵), ℝ*, < ) ≤ 𝐶 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶))
74, 5, 6syl2anc 585 . 2 (𝜑 → (sup(ran (𝑥𝐴𝐵), ℝ*, < ) ≤ 𝐶 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶))
8 nfmpt1 5214 . . . . . . . 8 𝑥(𝑥𝐴𝐵)
98nfrn 5908 . . . . . . 7 𝑥ran (𝑥𝐴𝐵)
10 nfv 1918 . . . . . . 7 𝑥 𝑧𝐶
119, 10nfralw 3293 . . . . . 6 𝑥𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶
121, 11nfan 1903 . . . . 5 𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶)
13 simpr 486 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
142elrnmpt1 5914 . . . . . . . . 9 ((𝑥𝐴𝐵 ∈ ℝ*) → 𝐵 ∈ ran (𝑥𝐴𝐵))
1513, 3, 14syl2anc 585 . . . . . . . 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 5109 . . . . . . . 8 (𝑧 = 𝐵 → (𝑧𝐶𝐵𝐶))
1918rspcva 3578 . . . . . . 7 ((𝐵 ∈ ran (𝑥𝐴𝐵) ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶) → 𝐵𝐶)
2016, 17, 19syl2anc 585 . . . . . 6 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶) ∧ 𝑥𝐴) → 𝐵𝐶)
2120ex 414 . . . . 5 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶) → (𝑥𝐴𝐵𝐶))
2212, 21ralrimi 3239 . . . 4 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶) → ∀𝑥𝐴 𝐵𝐶)
2322ex 414 . . 3 (𝜑 → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶 → ∀𝑥𝐴 𝐵𝐶))
24 vex 3448 . . . . . . . . 9 𝑧 ∈ V
252elrnmpt 5912 . . . . . . . . 9 (𝑧 ∈ V → (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵))
2624, 25ax-mp 5 . . . . . . . 8 (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵)
2726biimpi 215 . . . . . . 7 (𝑧 ∈ ran (𝑥𝐴𝐵) → ∃𝑥𝐴 𝑧 = 𝐵)
2827adantl 483 . . . . . 6 ((∀𝑥𝐴 𝐵𝐶𝑧 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 𝑧 = 𝐵)
29 nfra1 3266 . . . . . . . 8 𝑥𝑥𝐴 𝐵𝐶
30 rspa 3230 . . . . . . . . . 10 ((∀𝑥𝐴 𝐵𝐶𝑥𝐴) → 𝐵𝐶)
3118biimprcd 250 . . . . . . . . . 10 (𝐵𝐶 → (𝑧 = 𝐵𝑧𝐶))
3230, 31syl 17 . . . . . . . . 9 ((∀𝑥𝐴 𝐵𝐶𝑥𝐴) → (𝑧 = 𝐵𝑧𝐶))
3332ex 414 . . . . . . . 8 (∀𝑥𝐴 𝐵𝐶 → (𝑥𝐴 → (𝑧 = 𝐵𝑧𝐶)))
3429, 10, 33rexlimd 3248 . . . . . . 7 (∀𝑥𝐴 𝐵𝐶 → (∃𝑥𝐴 𝑧 = 𝐵𝑧𝐶))
3534adantr 482 . . . . . 6 ((∀𝑥𝐴 𝐵𝐶𝑧 ∈ ran (𝑥𝐴𝐵)) → (∃𝑥𝐴 𝑧 = 𝐵𝑧𝐶))
3628, 35mpd 15 . . . . 5 ((∀𝑥𝐴 𝐵𝐶𝑧 ∈ ran (𝑥𝐴𝐵)) → 𝑧𝐶)
3736ralrimiva 3140 . . . 4 (∀𝑥𝐴 𝐵𝐶 → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶)
3837a1i 11 . . 3 (𝜑 → (∀𝑥𝐴 𝐵𝐶 → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶))
3923, 38impbid 211 . 2 (𝜑 → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝐶 ↔ ∀𝑥𝐴 𝐵𝐶))
407, 39bitrd 279 1 (𝜑 → (sup(ran (𝑥𝐴𝐵), ℝ*, < ) ≤ 𝐶 ↔ ∀𝑥𝐴 𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wnf 1786  wcel 2107  wral 3061  wrex 3070  Vcvv 3444  wss 3911   class class class wbr 5106  cmpt 5189  ran crn 5635  supcsup 9381  *cxr 11193   < clt 11194  cle 11195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-po 5546  df-so 5547  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-sup 9383  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393
This theorem is referenced by:  supxrleubrnmptf  43772
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