Theorem suprnmpt 38350

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 2949	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ ℝ)
4		suprnmpt.f	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
5	4	rnmptss 6299	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ ℝ → ran 𝐹 ⊆ ℝ)
6	3, 5	syl 17	. . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
7		nfv 1830	. . . . 5 ⊢ Ⅎ𝑥𝜑
8		nfmpt1 4675	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
9	4, 8	nfcxfr 2749	. . . . . . 7 ⊢ Ⅎ𝑥𝐹
10	9	nfrn 5289	. . . . . 6 ⊢ Ⅎ𝑥ran 𝐹
11		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑥∅
12	10, 11	nfne 2882	. . . . 5 ⊢ Ⅎ𝑥ran 𝐹 ≠ ∅
13		suprnmpt.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ ∅)
14		n0 3890	. . . . . 6 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
15	13, 14	sylib 207	. . . . 5 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
16		simpr 476	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
17	4	elrnmpt1 5295	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ran 𝐹)
18	16, 2, 17	syl2anc 691	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran 𝐹)
19		ne0i 3880	. . . . . 6 ⊢ (𝐵 ∈ ran 𝐹 → ran 𝐹 ≠ ∅)
20	18, 19	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran 𝐹 ≠ ∅)
21	7, 12, 15, 20	exlimdd 2075	. . . 4 ⊢ (𝜑 → ran 𝐹 ≠ ∅)
22		suprnmpt.bnd	. . . . 5 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
23		nfv 1830	. . . . . 6 ⊢ Ⅎ𝑦𝜑
24		nfre1 2988	. . . . . 6 ⊢ Ⅎ𝑦∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦
25		simp2 1055	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) → 𝑦 ∈ ℝ)
26		simpl1 1057	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ∧ 𝑧 ∈ ran 𝐹) → 𝜑)
27		simpl3 1059	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ∧ 𝑧 ∈ ran 𝐹) → ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
28		vex 3176	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
29	4	elrnmpt 5293	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ V → (𝑧 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
30	28, 29	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
31	30	biimpi 205	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ran 𝐹 → ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
32	31	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ∧ 𝑧 ∈ ran 𝐹) → ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
33		simp3 1056	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
34		nfra1 2925	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦
35		nfre1 2988	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝑧 = 𝐵
36	7, 34, 35	nf3an 1819	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
37		nfv 1830	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝑧 ≤ 𝑦
38		simp3 1056	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝑧 = 𝐵)
39		rspa 2914	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝑦)
40	39	3adant3 1074	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝐵 ≤ 𝑦)
41	38, 40	eqbrtrd 4605	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝑧 ≤ 𝑦)
42	41	3exp 1256	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 → (𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝑧 ≤ 𝑦)))
43	42	3ad2ant2 1076	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → (𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝑧 ≤ 𝑦)))
44	36, 37, 43	rexlimd 3008	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ≤ 𝑦))
45	33, 44	mpd 15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → 𝑧 ≤ 𝑦)
46	26, 27, 32, 45	syl3anc 1318	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ∧ 𝑧 ∈ ran 𝐹) → 𝑧 ≤ 𝑦)
47	46	ralrimiva 2949	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) → ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦)
48		19.8a 2039	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦) → ∃𝑦(𝑦 ∈ ℝ ∧ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦))
49	25, 47, 48	syl2anc 691	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) → ∃𝑦(𝑦 ∈ ℝ ∧ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦))
50		df-rex 2902	. . . . . . . 8 ⊢ (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦 ↔ ∃𝑦(𝑦 ∈ ℝ ∧ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦))
51	49, 50	sylibr 223	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦)
52	51	3exp 1256	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ ℝ → (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦)))
53	23, 24, 52	rexlimd 3008	. . . . 5 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦))
54	22, 53	mpd 15	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦)
55		suprcl 10862	. . . 4 ⊢ ((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
56	6, 21, 54, 55	syl3anc 1318	. . 3 ⊢ (𝜑 → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
57	1, 56	syl5eqel 2692	. 2 ⊢ (𝜑 → 𝐶 ∈ ℝ)
58	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran 𝐹 ⊆ ℝ)
59	54	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦)
60		suprub 10863	. . . . 5 ⊢ (((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦) ∧ 𝐵 ∈ ran 𝐹) → 𝐵 ≤ sup(ran 𝐹, ℝ, < ))
61	58, 20, 59, 18, 60	syl31anc 1321	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ sup(ran 𝐹, ℝ, < ))
62	61, 1	syl6breqr 4625	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝐶)
63	62	ralrimiva 2949	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶)
64	57, 63	jca 553	1 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∅c0 3874   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039  supcsup 8229  ℝcr 9814   < clt 9953   ≤ cle 9954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148

This theorem is referenced by:  ioodvbdlimc1lem1  38821  ioodvbdlimc1lem2  38822  ioodvbdlimc2lem  38824