Theorem suprleubrnmpt 46001

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 2764	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3		suprleubrnmpt.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
4	1, 2, 3	rnmptssd 7107	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ)
5		suprleubrnmpt.a	. . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅)
6	1, 3, 2, 5	rnmptn0 6233	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅)
7		suprleubrnmpt.e	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
8	1, 7	rnmptbdd 45825	. . 3 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑦)
9		suprleubrnmpt.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ)
10		suprleub 12160	. . 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 1394	. 2 ⊢ (𝜑 → (sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) ≤ 𝐶 ↔ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝐶))
12		nfmpt1 5201	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
13	12	nfrn 5930	. . . . . . 7 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
14		nfv 1936	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧 ≤ 𝐶
15	13, 14	nfralw 3311	. . . . . 6 ⊢ Ⅎ𝑥∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝐶
16	1, 15	nfan 1921	. . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝐶)
17		simpr 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
18	2	elrnmpt1 5938	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
19	17, 3, 18	syl2anc 593	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
20	19	adantlr 725	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝐶) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
21		simplr 778	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝐶) ∧ 𝑥 ∈ 𝐴) → ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝐶)
22		breq1 5105	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝑧 ≤ 𝐶 ↔ 𝐵 ≤ 𝐶))
23	22	rspcva 3581	. . . . . . 7 ⊢ ((𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝐶) → 𝐵 ≤ 𝐶)
24	20, 21, 23	syl2anc 593	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝐶) ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝐶)
25	24	ex 416	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝐶) → (𝑥 ∈ 𝐴 → 𝐵 ≤ 𝐶))
26	16, 25	ralrimi 3262	. . . 4 ⊢ ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝐶) → ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶)
27	26	ex 416	. . 3 ⊢ (𝜑 → (∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝐶 → ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶))
28		vex 3460	. . . . . . . 8 ⊢ 𝑧 ∈ V
29	2	elrnmpt 5936	. . . . . . . 8 ⊢ (𝑧 ∈ V → (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
30	28, 29	ax-mp 5	. . . . . . 7 ⊢ (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
31	30	bilani 508	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶 ∧ 𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
32		nfra1 3288	. . . . . . . 8 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶
33		rspa 3253	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝐶)
34	22	biimprcd 252	. . . . . . . . . 10 ⊢ (𝐵 ≤ 𝐶 → (𝑧 = 𝐵 → 𝑧 ≤ 𝐶))
35	33, 34	syl 17	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶 ∧ 𝑥 ∈ 𝐴) → (𝑧 = 𝐵 → 𝑧 ≤ 𝐶))
36	35	ex 416	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶 → (𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝑧 ≤ 𝐶)))
37	32, 14, 36	rexlimd 3271	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶 → (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ≤ 𝐶))
38	37	adantr 484	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶 ∧ 𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ≤ 𝐶))
39	31, 38	mpd 15	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶 ∧ 𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑧 ≤ 𝐶)
40	39	ralrimiva 3156	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶 → ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝐶)
41	40	a1i 11	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶 → ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝐶))
42	27, 41	impbid 214	. 2 ⊢ (𝜑 → (∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶))
43	11, 42	bitrd 281	1 ⊢ (𝜑 → (sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) ≤ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562  Ⅎwnf 1805   ∈ wcel 2144   ≠ wne 2959  ∀wral 3078  ∃wrex 3088  Vcvv 3456   ⊆ wss 3906  ∅c0 4287   class class class wbr 5102   ↦ cmpt 5183  ran crn 5650  supcsup 9388  ℝcr 11074   < clt 11218   ≤ cle 11219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-po 5557  df-so 5558  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-er 8680  df-en 8930  df-dom 8931  df-sdom 8932  df-sup 9390  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419

This theorem is referenced by:  smfsuplem1  47390