Theorem fin23lem36 9821

Description: Lemma for fin23 9862. Weak order property of 𝑌. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Hypotheses

Ref	Expression
fin23lem33.f	⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
fin23lem.f	⊢ (𝜑 → ℎ:ω–1-1→V)
fin23lem.g	⊢ (𝜑 → ∪ ran ℎ ⊆ 𝐺)
fin23lem.h	⊢ (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) → ((𝑖‘𝑗):ω–1-1→V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗)))
fin23lem.i	⊢ 𝑌 = (rec(𝑖, ℎ) ↾ ω)

Assertion

Ref	Expression
fin23lem36	⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵 ⊆ 𝐴 ∧ 𝜑)) → ∪ ran (𝑌‘𝐴) ⊆ ∪ ran (𝑌‘𝐵))

Distinct variable groups:   𝑔,𝑎,𝑖,𝑗,𝑥   𝐴,𝑎,𝑗   ℎ,𝑎,𝐺,𝑔,𝑖,𝑗,𝑥   𝐵,𝑎   𝐹,𝑎   𝜑,𝑎,𝑗   𝑌,𝑎,𝑗

Allowed substitution hints:   𝜑(𝑥,𝑔,ℎ,𝑖)   𝐴(𝑥,𝑔,ℎ,𝑖)   𝐵(𝑥,𝑔,ℎ,𝑖,𝑗)   𝐹(𝑥,𝑔,ℎ,𝑖,𝑗)   𝑌(𝑥,𝑔,ℎ,𝑖)

Proof of Theorem fin23lem36

Dummy variable 𝑏 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6663	. . . . . . 7 ⊢ (𝑎 = 𝐵 → (𝑌‘𝑎) = (𝑌‘𝐵))
2	1	rneqd 5784	. . . . . 6 ⊢ (𝑎 = 𝐵 → ran (𝑌‘𝑎) = ran (𝑌‘𝐵))
3	2	unieqd 4815	. . . . 5 ⊢ (𝑎 = 𝐵 → ∪ ran (𝑌‘𝑎) = ∪ ran (𝑌‘𝐵))
4	3	sseq1d 3925	. . . 4 ⊢ (𝑎 = 𝐵 → (∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵) ↔ ∪ ran (𝑌‘𝐵) ⊆ ∪ ran (𝑌‘𝐵)))
5	4	imbi2d 344	. . 3 ⊢ (𝑎 = 𝐵 → ((𝜑 → ∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵)) ↔ (𝜑 → ∪ ran (𝑌‘𝐵) ⊆ ∪ ran (𝑌‘𝐵))))
6		fveq2 6663	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑌‘𝑎) = (𝑌‘𝑏))
7	6	rneqd 5784	. . . . . 6 ⊢ (𝑎 = 𝑏 → ran (𝑌‘𝑎) = ran (𝑌‘𝑏))
8	7	unieqd 4815	. . . . 5 ⊢ (𝑎 = 𝑏 → ∪ ran (𝑌‘𝑎) = ∪ ran (𝑌‘𝑏))
9	8	sseq1d 3925	. . . 4 ⊢ (𝑎 = 𝑏 → (∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵) ↔ ∪ ran (𝑌‘𝑏) ⊆ ∪ ran (𝑌‘𝐵)))
10	9	imbi2d 344	. . 3 ⊢ (𝑎 = 𝑏 → ((𝜑 → ∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵)) ↔ (𝜑 → ∪ ran (𝑌‘𝑏) ⊆ ∪ ran (𝑌‘𝐵))))
11		fveq2 6663	. . . . . . 7 ⊢ (𝑎 = suc 𝑏 → (𝑌‘𝑎) = (𝑌‘suc 𝑏))
12	11	rneqd 5784	. . . . . 6 ⊢ (𝑎 = suc 𝑏 → ran (𝑌‘𝑎) = ran (𝑌‘suc 𝑏))
13	12	unieqd 4815	. . . . 5 ⊢ (𝑎 = suc 𝑏 → ∪ ran (𝑌‘𝑎) = ∪ ran (𝑌‘suc 𝑏))
14	13	sseq1d 3925	. . . 4 ⊢ (𝑎 = suc 𝑏 → (∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵) ↔ ∪ ran (𝑌‘suc 𝑏) ⊆ ∪ ran (𝑌‘𝐵)))
15	14	imbi2d 344	. . 3 ⊢ (𝑎 = suc 𝑏 → ((𝜑 → ∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵)) ↔ (𝜑 → ∪ ran (𝑌‘suc 𝑏) ⊆ ∪ ran (𝑌‘𝐵))))
16		fveq2 6663	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑌‘𝑎) = (𝑌‘𝐴))
17	16	rneqd 5784	. . . . . 6 ⊢ (𝑎 = 𝐴 → ran (𝑌‘𝑎) = ran (𝑌‘𝐴))
18	17	unieqd 4815	. . . . 5 ⊢ (𝑎 = 𝐴 → ∪ ran (𝑌‘𝑎) = ∪ ran (𝑌‘𝐴))
19	18	sseq1d 3925	. . . 4 ⊢ (𝑎 = 𝐴 → (∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵) ↔ ∪ ran (𝑌‘𝐴) ⊆ ∪ ran (𝑌‘𝐵)))
20	19	imbi2d 344	. . 3 ⊢ (𝑎 = 𝐴 → ((𝜑 → ∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵)) ↔ (𝜑 → ∪ ran (𝑌‘𝐴) ⊆ ∪ ran (𝑌‘𝐵))))
21		ssid 3916	. . . 4 ⊢ ∪ ran (𝑌‘𝐵) ⊆ ∪ ran (𝑌‘𝐵)
22	21	2a1i 12	. . 3 ⊢ (𝐵 ∈ ω → (𝜑 → ∪ ran (𝑌‘𝐵) ⊆ ∪ ran (𝑌‘𝐵)))
23		simprr 772	. . . . . . . 8 ⊢ (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵 ⊆ 𝑏 ∧ 𝜑)) → 𝜑)
24		simpll 766	. . . . . . . 8 ⊢ (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵 ⊆ 𝑏 ∧ 𝜑)) → 𝑏 ∈ ω)
25		fin23lem33.f	. . . . . . . . 9 ⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
26		fin23lem.f	. . . . . . . . 9 ⊢ (𝜑 → ℎ:ω–1-1→V)
27		fin23lem.g	. . . . . . . . 9 ⊢ (𝜑 → ∪ ran ℎ ⊆ 𝐺)
28		fin23lem.h	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) → ((𝑖‘𝑗):ω–1-1→V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗)))
29		fin23lem.i	. . . . . . . . 9 ⊢ 𝑌 = (rec(𝑖, ℎ) ↾ ω)
30	25, 26, 27, 28, 29	fin23lem35 9820	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ ω) → ∪ ran (𝑌‘suc 𝑏) ⊊ ∪ ran (𝑌‘𝑏))
31	23, 24, 30	syl2anc 587	. . . . . . 7 ⊢ (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵 ⊆ 𝑏 ∧ 𝜑)) → ∪ ran (𝑌‘suc 𝑏) ⊊ ∪ ran (𝑌‘𝑏))
32	31	pssssd 4005	. . . . . 6 ⊢ (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵 ⊆ 𝑏 ∧ 𝜑)) → ∪ ran (𝑌‘suc 𝑏) ⊆ ∪ ran (𝑌‘𝑏))
33		sstr2 3901	. . . . . 6 ⊢ (∪ ran (𝑌‘suc 𝑏) ⊆ ∪ ran (𝑌‘𝑏) → (∪ ran (𝑌‘𝑏) ⊆ ∪ ran (𝑌‘𝐵) → ∪ ran (𝑌‘suc 𝑏) ⊆ ∪ ran (𝑌‘𝐵)))
34	32, 33	syl 17	. . . . 5 ⊢ (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵 ⊆ 𝑏 ∧ 𝜑)) → (∪ ran (𝑌‘𝑏) ⊆ ∪ ran (𝑌‘𝐵) → ∪ ran (𝑌‘suc 𝑏) ⊆ ∪ ran (𝑌‘𝐵)))
35	34	expr 460	. . . 4 ⊢ (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑏) → (𝜑 → (∪ ran (𝑌‘𝑏) ⊆ ∪ ran (𝑌‘𝐵) → ∪ ran (𝑌‘suc 𝑏) ⊆ ∪ ran (𝑌‘𝐵))))
36	35	a2d 29	. . 3 ⊢ (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑏) → ((𝜑 → ∪ ran (𝑌‘𝑏) ⊆ ∪ ran (𝑌‘𝐵)) → (𝜑 → ∪ ran (𝑌‘suc 𝑏) ⊆ ∪ ran (𝑌‘𝐵))))
37	5, 10, 15, 20, 22, 36	findsg 7615	. 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝐴) → (𝜑 → ∪ ran (𝑌‘𝐴) ⊆ ∪ ran (𝑌‘𝐵)))
38	37	impr 458	1 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵 ⊆ 𝐴 ∧ 𝜑)) → ∪ ran (𝑌‘𝐴) ⊆ ∪ ran (𝑌‘𝐵))

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111  {cab 2735  ∀wral 3070  Vcvv 3409   ⊆ wss 3860   ⊊ wpss 3861  𝒫 cpw 4497  ∪ cuni 4801  ∩ cint 4841  ran crn 5529   ↾ cres 5530  suc csuc 6176  –1-1→wf1 6337  ‘cfv 6340  (class class class)co 7156  ωcom 7585  reccrdg 8061   ↑_m cmap 8422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-un 7465

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-om 7586  df-wrecs 7963  df-recs 8024  df-rdg 8062

This theorem is referenced by: (None)