Theorem dstfrvunirn 30536

Description: The limit of all preimage maps by the "lower than or equal" relation is the universe. (Contributed by Thierry Arnoux, 12-Feb-2017.)

Hypotheses

Ref	Expression
dstfrv.1	⊢ (𝜑 → 𝑃 ∈ Prob)
dstfrv.2	⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))

Assertion

Ref	Expression
dstfrvunirn	⊢ (𝜑 → ∪ ran (𝑛 ∈ ℕ ↦ (𝑋∘RV/𝑐 ≤ 𝑛)) = ∪ dom 𝑃)

Distinct variable groups:   𝑃,𝑛   𝑛,𝑋   𝜑,𝑛

Proof of Theorem dstfrvunirn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1red 10055	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ dom 𝑃) → 1 ∈ ℝ)
2		dstfrv.1	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ Prob)
3		dstfrv.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))
4	2, 3	rrvvf 30506	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋:∪ dom 𝑃⟶ℝ)
5	4	ffvelrnda 6359	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ dom 𝑃) → (𝑋‘𝑥) ∈ ℝ)
6	1, 5	ifcld 4131	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ dom 𝑃) → if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥)) ∈ ℝ)
7		breq2 4657	. . . . . . . . . 10 ⊢ (1 = if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥)) → (1 ≤ 1 ↔ 1 ≤ if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥))))
8		breq2 4657	. . . . . . . . . 10 ⊢ ((𝑋‘𝑥) = if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥)) → (1 ≤ (𝑋‘𝑥) ↔ 1 ≤ if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥))))
9		1le1 10655	. . . . . . . . . . 11 ⊢ 1 ≤ 1
10	9	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ∪ dom 𝑃) ∧ (𝑋‘𝑥) < 1) → 1 ≤ 1)
11	1, 5	lenltd 10183	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ dom 𝑃) → (1 ≤ (𝑋‘𝑥) ↔ ¬ (𝑋‘𝑥) < 1))
12	11	biimpar 502	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ∪ dom 𝑃) ∧ ¬ (𝑋‘𝑥) < 1) → 1 ≤ (𝑋‘𝑥))
13	7, 8, 10, 12	ifbothda 4123	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ dom 𝑃) → 1 ≤ if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥)))
14		flge1nn 12622	. . . . . . . . 9 ⊢ ((if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥)) ∈ ℝ ∧ 1 ≤ if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥))) → (⌊‘if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥))) ∈ ℕ)
15	6, 13, 14	syl2anc 693	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ dom 𝑃) → (⌊‘if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥))) ∈ ℕ)
16	15	peano2nnd 11037	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ dom 𝑃) → ((⌊‘if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥))) + 1) ∈ ℕ)
17	2	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ dom 𝑃) → 𝑃 ∈ Prob)
18	3	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ dom 𝑃) → 𝑋 ∈ (rRndVar‘𝑃))
19	16	nnred 11035	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ dom 𝑃) → ((⌊‘if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥))) + 1) ∈ ℝ)
20		simpr 477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ dom 𝑃) → 𝑥 ∈ ∪ dom 𝑃)
21		breq2 4657	. . . . . . . . . 10 ⊢ (1 = if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥)) → ((𝑋‘𝑥) ≤ 1 ↔ (𝑋‘𝑥) ≤ if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥))))
22		breq2 4657	. . . . . . . . . 10 ⊢ ((𝑋‘𝑥) = if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥)) → ((𝑋‘𝑥) ≤ (𝑋‘𝑥) ↔ (𝑋‘𝑥) ≤ if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥))))
23	5	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ∪ dom 𝑃) ∧ (𝑋‘𝑥) < 1) → (𝑋‘𝑥) ∈ ℝ)
24		1red 10055	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ∪ dom 𝑃) ∧ (𝑋‘𝑥) < 1) → 1 ∈ ℝ)
25		simpr 477	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ∪ dom 𝑃) ∧ (𝑋‘𝑥) < 1) → (𝑋‘𝑥) < 1)
26	23, 24, 25	ltled 10185	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ∪ dom 𝑃) ∧ (𝑋‘𝑥) < 1) → (𝑋‘𝑥) ≤ 1)
27	5	leidd 10594	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ dom 𝑃) → (𝑋‘𝑥) ≤ (𝑋‘𝑥))
28	27	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ∪ dom 𝑃) ∧ ¬ (𝑋‘𝑥) < 1) → (𝑋‘𝑥) ≤ (𝑋‘𝑥))
29	21, 22, 26, 28	ifbothda 4123	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ dom 𝑃) → (𝑋‘𝑥) ≤ if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥)))
30		fllep1 12602	. . . . . . . . . 10 ⊢ (if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥)) ∈ ℝ → if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥)) ≤ ((⌊‘if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥))) + 1))
31	6, 30	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ dom 𝑃) → if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥)) ≤ ((⌊‘if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥))) + 1))
32	5, 6, 19, 29, 31	letrd 10194	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ dom 𝑃) → (𝑋‘𝑥) ≤ ((⌊‘if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥))) + 1))
33	17, 18, 19, 20, 32	dstfrvel 30535	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ dom 𝑃) → 𝑥 ∈ (𝑋∘RV/𝑐 ≤ ((⌊‘if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥))) + 1)))
34		oveq2 6658	. . . . . . . . 9 ⊢ (𝑛 = ((⌊‘if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥))) + 1) → (𝑋∘RV/𝑐 ≤ 𝑛) = (𝑋∘RV/𝑐 ≤ ((⌊‘if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥))) + 1)))
35	34	eleq2d 2687	. . . . . . . 8 ⊢ (𝑛 = ((⌊‘if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥))) + 1) → (𝑥 ∈ (𝑋∘RV/𝑐 ≤ 𝑛) ↔ 𝑥 ∈ (𝑋∘RV/𝑐 ≤ ((⌊‘if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥))) + 1))))
36	35	rspcev 3309	. . . . . . 7 ⊢ ((((⌊‘if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥))) + 1) ∈ ℕ ∧ 𝑥 ∈ (𝑋∘RV/𝑐 ≤ ((⌊‘if((𝑋‘𝑥) < 1, 1, (𝑋‘𝑥))) + 1))) → ∃𝑛 ∈ ℕ 𝑥 ∈ (𝑋∘RV/𝑐 ≤ 𝑛))
37	16, 33, 36	syl2anc 693	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ dom 𝑃) → ∃𝑛 ∈ ℕ 𝑥 ∈ (𝑋∘RV/𝑐 ≤ 𝑛))
38	37	ex 450	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ∪ dom 𝑃 → ∃𝑛 ∈ ℕ 𝑥 ∈ (𝑋∘RV/𝑐 ≤ 𝑛)))
39	2	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ Prob)
40	3	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ (rRndVar‘𝑃))
41		simpr 477	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
42	41	nnred 11035	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
43	39, 40, 42	orvclteel 30534	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑋∘RV/𝑐 ≤ 𝑛) ∈ dom 𝑃)
44		elunii 4441	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝑋∘RV/𝑐 ≤ 𝑛) ∧ (𝑋∘RV/𝑐 ≤ 𝑛) ∈ dom 𝑃) → 𝑥 ∈ ∪ dom 𝑃)
45	44	expcom 451	. . . . . . 7 ⊢ ((𝑋∘RV/𝑐 ≤ 𝑛) ∈ dom 𝑃 → (𝑥 ∈ (𝑋∘RV/𝑐 ≤ 𝑛) → 𝑥 ∈ ∪ dom 𝑃))
46	43, 45	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑥 ∈ (𝑋∘RV/𝑐 ≤ 𝑛) → 𝑥 ∈ ∪ dom 𝑃))
47	46	rexlimdva 3031	. . . . 5 ⊢ (𝜑 → (∃𝑛 ∈ ℕ 𝑥 ∈ (𝑋∘RV/𝑐 ≤ 𝑛) → 𝑥 ∈ ∪ dom 𝑃))
48	38, 47	impbid 202	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ∪ dom 𝑃 ↔ ∃𝑛 ∈ ℕ 𝑥 ∈ (𝑋∘RV/𝑐 ≤ 𝑛)))
49		eliun 4524	. . . 4 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ ℕ (𝑋∘RV/𝑐 ≤ 𝑛) ↔ ∃𝑛 ∈ ℕ 𝑥 ∈ (𝑋∘RV/𝑐 ≤ 𝑛))
50	48, 49	syl6bbr 278	. . 3 ⊢ (𝜑 → (𝑥 ∈ ∪ dom 𝑃 ↔ 𝑥 ∈ ∪ 𝑛 ∈ ℕ (𝑋∘RV/𝑐 ≤ 𝑛)))
51	50	eqrdv 2620	. 2 ⊢ (𝜑 → ∪ dom 𝑃 = ∪ 𝑛 ∈ ℕ (𝑋∘RV/𝑐 ≤ 𝑛))
52		ovex 6678	. . 3 ⊢ (𝑋∘RV/𝑐 ≤ 𝑛) ∈ V
53	52	dfiun3 5380	. 2 ⊢ ∪ 𝑛 ∈ ℕ (𝑋∘RV/𝑐 ≤ 𝑛) = ∪ ran (𝑛 ∈ ℕ ↦ (𝑋∘RV/𝑐 ≤ 𝑛))
54	51, 53	syl6req 2673	1 ⊢ (𝜑 → ∪ ran (𝑛 ∈ ℕ ↦ (𝑋∘RV/𝑐 ≤ 𝑛)) = ∪ dom 𝑃)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  ifcif 4086  ∪ cuni 4436  ∪ ciun 4520   class class class wbr 4653   ↦ cmpt 4729  dom cdm 5114  ran crn 5115  ‘cfv 5888  (class class class)co 6650  ℝcr 9935  1c1 9937   + caddc 9939   < clt 10074   ≤ cle 10075  ℕcn 11020  ⌊cfl 12591  Probcprb 30469  rRndVarcrrv 30502  ∘_RV/𝑐corvc 30517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-ac2 9285  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-ac 8939  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-ioo 12179  df-ioc 12180  df-fl 12593  df-topgen 16104  df-top 20699  df-bases 20750  df-cld 20823  df-esum 30090  df-siga 30171  df-sigagen 30202  df-brsiga 30245  df-meas 30259  df-mbfm 30313  df-prob 30470  df-rrv 30503  df-orvc 30518

This theorem is referenced by:  dstfrvclim1  30539