Theorem suprnmpt 45749

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 3154	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ ℝ)
4		suprnmpt.f	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
5	4	rnmptss 7104	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ ℝ → ran 𝐹 ⊆ ℝ)
6	3, 5	syl 17	. . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
7		nfv 1934	. . . . 5 ⊢ Ⅎ𝑥𝜑
8		suprnmpt.a	. . . . 5 ⊢ (𝜑 → 𝐴 ≠ ∅)
9	7, 2, 4, 8	rnmptn0 6231	. . . 4 ⊢ (𝜑 → ran 𝐹 ≠ ∅)
10		suprnmpt.bnd	. . . . 5 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
11		nfv 1934	. . . . . 6 ⊢ Ⅎ𝑦𝜑
12		nfre1 3287	. . . . . 6 ⊢ Ⅎ𝑦∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦
13		simp2 1150	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) → 𝑦 ∈ ℝ)
14		simpl1 1205	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ∧ 𝑧 ∈ ran 𝐹) → 𝜑)
15		simpl3 1207	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ∧ 𝑧 ∈ ran 𝐹) → ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
16		vex 3458	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
17	4	elrnmpt 5934	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ V → (𝑧 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
18	16, 17	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
19	18	bilani 508	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ∧ 𝑧 ∈ ran 𝐹) → ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
20		simp3 1151	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
21		nfra1 3286	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦
22		nfre1 3287	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝑧 = 𝐵
23	7, 21, 22	nf3an 1921	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
24		nfv 1934	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝑧 ≤ 𝑦
25		simp3 1151	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝑧 = 𝐵)
26		rspa 3251	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝑦)
27	26	3adant3 1145	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝐵 ≤ 𝑦)
28	25, 27	eqbrtrd 5122	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝑧 ≤ 𝑦)
29	28	3exp 1132	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 → (𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝑧 ≤ 𝑦)))
30	29	3ad2ant2 1147	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → (𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝑧 ≤ 𝑦)))
31	23, 24, 30	rexlimd 3269	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ≤ 𝑦))
32	20, 31	mpd 15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → 𝑧 ≤ 𝑦)
33	14, 15, 19, 32	syl3anc 1390	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ∧ 𝑧 ∈ ran 𝐹) → 𝑧 ≤ 𝑦)
34	33	ralrimiva 3154	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) → ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦)
35		19.8a 2216	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦) → ∃𝑦(𝑦 ∈ ℝ ∧ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦))
36	13, 34, 35	syl2anc 593	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) → ∃𝑦(𝑦 ∈ ℝ ∧ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦))
37		df-rex 3087	. . . . . . . 8 ⊢ (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦 ↔ ∃𝑦(𝑦 ∈ ℝ ∧ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦))
38	36, 37	sylibr 236	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦)
39	38	3exp 1132	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ ℝ → (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦)))
40	11, 12, 39	rexlimd 3269	. . . . 5 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦))
41	10, 40	mpd 15	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦)
42		suprcl 12152	. . . 4 ⊢ ((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
43	6, 9, 41, 42	syl3anc 1390	. . 3 ⊢ (𝜑 → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
44	1, 43	eqeltrid 2866	. 2 ⊢ (𝜑 → 𝐶 ∈ ℝ)
45	6	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran 𝐹 ⊆ ℝ)
46		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
47	4	elrnmpt1 5936	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ran 𝐹)
48	46, 2, 47	syl2anc 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran 𝐹)
49	48	ne0d 4294	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran 𝐹 ≠ ∅)
50	41	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦)
51		suprub 12153	. . . . 5 ⊢ (((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦) ∧ 𝐵 ∈ ran 𝐹) → 𝐵 ≤ sup(ran 𝐹, ℝ, < ))
52	45, 49, 50, 48, 51	syl31anc 1392	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ sup(ran 𝐹, ℝ, < ))
53	52, 1	breqtrrdi 5142	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝐶)
54	53	ralrimiva 3154	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶)
55	44, 54	jca 519	1 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1098   = wceq 1560  ∃wex 1799   ∈ wcel 2142   ≠ wne 2957  ∀wral 3076  ∃wrex 3086  Vcvv 3454   ⊆ wss 3904  ∅c0 4285   class class class wbr 5100   ↦ cmpt 5181  ran crn 5648  supcsup 9386  ℝcr 11072   < clt 11216   ≤ cle 11217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150  ax-pre-sup 11151

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-po 5555  df-so 5556  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-er 8678  df-en 8928  df-dom 8929  df-sdom 8930  df-sup 9388  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417

This theorem is referenced by:  ioodvbdlimc1lem1  46502  ioodvbdlimc1lem2  46503  ioodvbdlimc2lem  46505