Theorem suprnmpt 43856

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 3147	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ ℝ)
4		suprnmpt.f	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
5	4	rnmptss 7119	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ ℝ → ran 𝐹 ⊆ ℝ)
6	3, 5	syl 17	. . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
7		nfv 1918	. . . . 5 ⊢ Ⅎ𝑥𝜑
8		suprnmpt.a	. . . . 5 ⊢ (𝜑 → 𝐴 ≠ ∅)
9	7, 2, 4, 8	rnmptn0 6241	. . . 4 ⊢ (𝜑 → ran 𝐹 ≠ ∅)
10		suprnmpt.bnd	. . . . 5 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
11		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑦𝜑
12		nfre1 3283	. . . . . 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 3479	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
17	4	elrnmpt 5954	. . . . . . . . . . . . . 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 215	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ran 𝐹 → ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
20	19	adantl 483	. . . . . . . . . . 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 3282	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦
23		nfre1 3283	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝑧 = 𝐵
24	7, 22, 23	nf3an 1905	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
25		nfv 1918	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝑧 ≤ 𝑦
26		simp3 1139	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝑧 = 𝐵)
27		rspa 3246	. . . . . . . . . . . . . . . . 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 5170	. . . . . . . . . . . . . . 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 3264	. . . . . . . . . . . 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 1372	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ∧ 𝑧 ∈ ran 𝐹) → 𝑧 ≤ 𝑦)
35	34	ralrimiva 3147	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) → ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦)
36		19.8a 2175	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦) → ∃𝑦(𝑦 ∈ ℝ ∧ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦))
37	13, 35, 36	syl2anc 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) → ∃𝑦(𝑦 ∈ ℝ ∧ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦))
38		df-rex 3072	. . . . . . . 8 ⊢ (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦 ↔ ∃𝑦(𝑦 ∈ ℝ ∧ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦))
39	37, 38	sylibr 233	. . . . . . 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 3264	. . . . 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 12171	. . . 4 ⊢ ((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
44	6, 9, 42, 43	syl3anc 1372	. . 3 ⊢ (𝜑 → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
45	1, 44	eqeltrid 2838	. 2 ⊢ (𝜑 → 𝐶 ∈ ℝ)
46	6	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran 𝐹 ⊆ ℝ)
47		simpr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
48	4	elrnmpt1 5956	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ran 𝐹)
49	47, 2, 48	syl2anc 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran 𝐹)
50	49	ne0d 4335	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran 𝐹 ≠ ∅)
51	42	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑦)
52		suprub 12172	. . . . 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 1374	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ sup(ran 𝐹, ℝ, < ))
54	53, 1	breqtrrdi 5190	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝐶)
55	54	ralrimiva 3147	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶)
56	45, 55	jca 513	1 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  ∃wex 1782   ∈ wcel 2107   ≠ wne 2941  ∀wral 3062  ∃wrex 3071  Vcvv 3475   ⊆ wss 3948  ∅c0 4322   class class class wbr 5148   ↦ cmpt 5231  ran crn 5677  supcsup 9432  ℝcr 11106   < clt 11245   ≤ cle 11246

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 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185

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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-po 5588  df-so 5589  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-sup 9434  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444

This theorem is referenced by:  ioodvbdlimc1lem1  44634  ioodvbdlimc1lem2  44635  ioodvbdlimc2lem  44637