Theorem suprleubrnmpt 45433

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	⊢ Ⅎ𝑥𝜑
suprleubrnmpt.a	⊢ (𝜑 → 𝐴 ≠ ∅)
suprleubrnmpt.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
suprleubrnmpt.e	⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
suprleubrnmpt.c	⊢ (𝜑 → 𝐶 ∈ ℝ)

Assertion

Ref	Expression
suprleubrnmpt	⊢ (𝜑 → (sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) ≤ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶))

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.

Step	Hyp	Ref	Expression
1		suprleubrnmpt.x	. . . 4 ⊢ Ⅎ𝑥𝜑
2		eqid 2737	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3		suprleubrnmpt.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
4	1, 2, 3	rnmptssd 45201	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ)
5		suprleubrnmpt.a	. . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅)
6	1, 3, 2, 5	rnmptn0 6264	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅)
7		suprleubrnmpt.e	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
8	1, 7	rnmptbdd 45252	. . 3 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑦)
9		suprleubrnmpt.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ)
10		suprleub 12234	. . 3 ⊢ (((ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ ∧ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑦) ∧ 𝐶 ∈ ℝ) → (sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) ≤ 𝐶 ↔ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝐶))
11	4, 6, 8, 9, 10	syl31anc 1375	. 2 ⊢ (𝜑 → (sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) ≤ 𝐶 ↔ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝐶))
12		nfmpt1 5250	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
13	12	nfrn 5963	. . . . . . 7 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
14		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧 ≤ 𝐶
15	13, 14	nfralw 3311	. . . . . 6 ⊢ Ⅎ𝑥∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝐶
16	1, 15	nfan 1899	. . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝐶)
17		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
18	2	elrnmpt1 5971	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
19	17, 3, 18	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
20	19	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝐶) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
21		simplr 769	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝐶) ∧ 𝑥 ∈ 𝐴) → ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝐶)
22		breq1 5146	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝑧 ≤ 𝐶 ↔ 𝐵 ≤ 𝐶))
23	22	rspcva 3620	. . . . . . 7 ⊢ ((𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝐶) → 𝐵 ≤ 𝐶)
24	20, 21, 23	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝐶) ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝐶)
25	24	ex 412	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝐶) → (𝑥 ∈ 𝐴 → 𝐵 ≤ 𝐶))
26	16, 25	ralrimi 3257	. . . 4 ⊢ ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝐶) → ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶)
27	26	ex 412	. . 3 ⊢ (𝜑 → (∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝐶 → ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶))
28		vex 3484	. . . . . . . . 9 ⊢ 𝑧 ∈ V
29	2	elrnmpt 5969	. . . . . . . . 9 ⊢ (𝑧 ∈ V → (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
30	28, 29	ax-mp 5	. . . . . . . 8 ⊢ (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
31	30	biimpi 216	. . . . . . 7 ⊢ (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
32	31	adantl 481	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶 ∧ 𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
33		nfra1 3284	. . . . . . . 8 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶
34		rspa 3248	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝐶)
35	22	biimprcd 250	. . . . . . . . . 10 ⊢ (𝐵 ≤ 𝐶 → (𝑧 = 𝐵 → 𝑧 ≤ 𝐶))
36	34, 35	syl 17	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶 ∧ 𝑥 ∈ 𝐴) → (𝑧 = 𝐵 → 𝑧 ≤ 𝐶))
37	36	ex 412	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶 → (𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝑧 ≤ 𝐶)))
38	33, 14, 37	rexlimd 3266	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶 → (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ≤ 𝐶))
39	38	adantr 480	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶 ∧ 𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ≤ 𝐶))
40	32, 39	mpd 15	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶 ∧ 𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑧 ≤ 𝐶)
41	40	ralrimiva 3146	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶 → ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝐶)
42	41	a1i 11	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶 → ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝐶))
43	27, 42	impbid 212	. 2 ⊢ (𝜑 → (∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶))
44	11, 43	bitrd 279	1 ⊢ (𝜑 → (sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) ≤ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  Vcvv 3480   ⊆ wss 3951  ∅c0 4333   class class class wbr 5143   ↦ cmpt 5225  ran crn 5686  supcsup 9480  ℝcr 11154   < clt 11295   ≤ cle 11296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495

This theorem is referenced by:  smfsuplem1  46826