Theorem suprnmpt 45179

Description: An explicit bound for the range of a bounded function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
suprnmpt.a	⊢ (𝜑 → 𝐴 ≠ ∅)
suprnmpt.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
suprnmpt.bnd	⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
suprnmpt.f	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
suprnmpt.c	⊢ 𝐶 = sup(ran 𝐹, ℝ, < )

Assertion

Ref	Expression
suprnmpt	⊢ (𝜑 → (𝐶 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶))

Distinct variable groups:   𝑥,𝐴   𝑦,𝐹   𝜑,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐹(𝑥)

Proof of Theorem suprnmpt

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		suprnmpt.c	. . 3 ⊢ 𝐶 = sup(ran 𝐹, ℝ, < )
2		suprnmpt.b	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
3	2	ralrimiva 3146	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ ℝ)
4		suprnmpt.f	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
5	4	rnmptss 7143	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ ℝ → ran 𝐹 ⊆ ℝ)
6	3, 5	syl 17	. . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
7		nfv 1914	. . . . 5 ⊢ Ⅎ𝑥𝜑
8		suprnmpt.a	. . . . 5 ⊢ (𝜑 → 𝐴 ≠ ∅)
9	7, 2, 4, 8	rnmptn0 6264	. . . 4 ⊢ (𝜑 → ran 𝐹 ≠ ∅)
10		suprnmpt.bnd	. . . . 5 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
11		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑦𝜑
12		nfre1 3285	. . . . . 6 ⊢ Ⅎ𝑦∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦
13		simp2 1138	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) → 𝑦 ∈ ℝ)
14		simpl1 1192	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ∧ 𝑧 ∈ ran 𝐹) → 𝜑)
15		simpl3 1194	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ∧ 𝑧 ∈ ran 𝐹) → ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
16		vex 3484	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
17	4	elrnmpt 5969	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ V → (𝑧 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
18	16, 17	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
19	18	biimpi 216	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ran 𝐹 → ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
20	19	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ∧ 𝑧 ∈ ran 𝐹) → ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
21		simp3 1139	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
22		nfra1 3284	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦
23		nfre1 3285	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝑧 = 𝐵
24	7, 22, 23	nf3an 1901	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
25		nfv 1914	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝑧 ≤ 𝑦
26		simp3 1139	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝑧 = 𝐵)
27		rspa 3248	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝑦)
28	27	3adant3 1133	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝐵 ≤ 𝑦)
29	26, 28	eqbrtrd 5165	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝑧 ≤ 𝑦)
30	29	3exp 1120	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 → (𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝑧 ≤ 𝑦)))
31	30	3ad2ant2 1135	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → (𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝑧 ≤ 𝑦)))
32	24, 25, 31	rexlimd 3266	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ≤ 𝑦))
33	21, 32	mpd 15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → 𝑧 ≤ 𝑦)
34	14, 15, 20, 33	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ∧ 𝑧 ∈ ran 𝐹) → 𝑧 ≤ 𝑦)
35	34	ralrimiva 3146	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) → ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦)
36		19.8a 2181	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦) → ∃𝑦(𝑦 ∈ ℝ ∧ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦))
37	13, 35, 36	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) → ∃𝑦(𝑦 ∈ ℝ ∧ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦))
38		df-rex 3071	. . . . . . . 8 ⊢ (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦 ↔ ∃𝑦(𝑦 ∈ ℝ ∧ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦))
39	37, 38	sylibr 234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦)
40	39	3exp 1120	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ ℝ → (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦)))
41	11, 12, 40	rexlimd 3266	. . . . 5 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦))
42	10, 41	mpd 15	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦)
43		suprcl 12228	. . . 4 ⊢ ((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
44	6, 9, 42, 43	syl3anc 1373	. . 3 ⊢ (𝜑 → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
45	1, 44	eqeltrid 2845	. 2 ⊢ (𝜑 → 𝐶 ∈ ℝ)
46	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran 𝐹 ⊆ ℝ)
47		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
48	4	elrnmpt1 5971	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ran 𝐹)
49	47, 2, 48	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran 𝐹)
50	49	ne0d 4342	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran 𝐹 ≠ ∅)
51	42	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦)
52		suprub 12229	. . . . 5 ⊢ (((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦) ∧ 𝐵 ∈ ran 𝐹) → 𝐵 ≤ sup(ran 𝐹, ℝ, < ))
53	46, 50, 51, 49, 52	syl31anc 1375	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ sup(ran 𝐹, ℝ, < ))
54	53, 1	breqtrrdi 5185	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝐶)
55	54	ralrimiva 3146	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶)
56	45, 55	jca 511	1 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540  ∃wex 1779   ∈ 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:  ioodvbdlimc1lem1  45946  ioodvbdlimc1lem2  45947  ioodvbdlimc2lem  45949