Theorem fin23lem36 10104

Description: Lemma for fin23 10145. 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 6774	. . . . . . 7 ⊢ (𝑎 = 𝐵 → (𝑌‘𝑎) = (𝑌‘𝐵))
2	1	rneqd 5847	. . . . . 6 ⊢ (𝑎 = 𝐵 → ran (𝑌‘𝑎) = ran (𝑌‘𝐵))
3	2	unieqd 4853	. . . . 5 ⊢ (𝑎 = 𝐵 → ∪ ran (𝑌‘𝑎) = ∪ ran (𝑌‘𝐵))
4	3	sseq1d 3952	. . . 4 ⊢ (𝑎 = 𝐵 → (∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵) ↔ ∪ ran (𝑌‘𝐵) ⊆ ∪ ran (𝑌‘𝐵)))
5	4	imbi2d 341	. . 3 ⊢ (𝑎 = 𝐵 → ((𝜑 → ∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵)) ↔ (𝜑 → ∪ ran (𝑌‘𝐵) ⊆ ∪ ran (𝑌‘𝐵))))
6		fveq2 6774	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑌‘𝑎) = (𝑌‘𝑏))
7	6	rneqd 5847	. . . . . 6 ⊢ (𝑎 = 𝑏 → ran (𝑌‘𝑎) = ran (𝑌‘𝑏))
8	7	unieqd 4853	. . . . 5 ⊢ (𝑎 = 𝑏 → ∪ ran (𝑌‘𝑎) = ∪ ran (𝑌‘𝑏))
9	8	sseq1d 3952	. . . 4 ⊢ (𝑎 = 𝑏 → (∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵) ↔ ∪ ran (𝑌‘𝑏) ⊆ ∪ ran (𝑌‘𝐵)))
10	9	imbi2d 341	. . 3 ⊢ (𝑎 = 𝑏 → ((𝜑 → ∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵)) ↔ (𝜑 → ∪ ran (𝑌‘𝑏) ⊆ ∪ ran (𝑌‘𝐵))))
11		fveq2 6774	. . . . . . 7 ⊢ (𝑎 = suc 𝑏 → (𝑌‘𝑎) = (𝑌‘suc 𝑏))
12	11	rneqd 5847	. . . . . 6 ⊢ (𝑎 = suc 𝑏 → ran (𝑌‘𝑎) = ran (𝑌‘suc 𝑏))
13	12	unieqd 4853	. . . . 5 ⊢ (𝑎 = suc 𝑏 → ∪ ran (𝑌‘𝑎) = ∪ ran (𝑌‘suc 𝑏))
14	13	sseq1d 3952	. . . 4 ⊢ (𝑎 = suc 𝑏 → (∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵) ↔ ∪ ran (𝑌‘suc 𝑏) ⊆ ∪ ran (𝑌‘𝐵)))
15	14	imbi2d 341	. . 3 ⊢ (𝑎 = suc 𝑏 → ((𝜑 → ∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵)) ↔ (𝜑 → ∪ ran (𝑌‘suc 𝑏) ⊆ ∪ ran (𝑌‘𝐵))))
16		fveq2 6774	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑌‘𝑎) = (𝑌‘𝐴))
17	16	rneqd 5847	. . . . . 6 ⊢ (𝑎 = 𝐴 → ran (𝑌‘𝑎) = ran (𝑌‘𝐴))
18	17	unieqd 4853	. . . . 5 ⊢ (𝑎 = 𝐴 → ∪ ran (𝑌‘𝑎) = ∪ ran (𝑌‘𝐴))
19	18	sseq1d 3952	. . . 4 ⊢ (𝑎 = 𝐴 → (∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵) ↔ ∪ ran (𝑌‘𝐴) ⊆ ∪ ran (𝑌‘𝐵)))
20	19	imbi2d 341	. . 3 ⊢ (𝑎 = 𝐴 → ((𝜑 → ∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵)) ↔ (𝜑 → ∪ ran (𝑌‘𝐴) ⊆ ∪ ran (𝑌‘𝐵))))
21		ssid 3943	. . . 4 ⊢ ∪ ran (𝑌‘𝐵) ⊆ ∪ ran (𝑌‘𝐵)
22	21	2a1i 12	. . 3 ⊢ (𝐵 ∈ ω → (𝜑 → ∪ ran (𝑌‘𝐵) ⊆ ∪ ran (𝑌‘𝐵)))
23		simprr 770	. . . . . . . 8 ⊢ (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵 ⊆ 𝑏 ∧ 𝜑)) → 𝜑)
24		simpll 764	. . . . . . . 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 10103	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ ω) → ∪ ran (𝑌‘suc 𝑏) ⊊ ∪ ran (𝑌‘𝑏))
31	23, 24, 30	syl2anc 584	. . . . . . 7 ⊢ (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵 ⊆ 𝑏 ∧ 𝜑)) → ∪ ran (𝑌‘suc 𝑏) ⊊ ∪ ran (𝑌‘𝑏))
32	31	pssssd 4032	. . . . . 6 ⊢ (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵 ⊆ 𝑏 ∧ 𝜑)) → ∪ ran (𝑌‘suc 𝑏) ⊆ ∪ ran (𝑌‘𝑏))
33		sstr2 3928	. . . . . 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 457	. . . 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 7746	. 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝐴) → (𝜑 → ∪ ran (𝑌‘𝐴) ⊆ ∪ ran (𝑌‘𝐵)))
38	37	impr 455	1 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵 ⊆ 𝐴 ∧ 𝜑)) → ∪ ran (𝑌‘𝐴) ⊆ ∪ ran (𝑌‘𝐵))

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1537   = wceq 1539   ∈ wcel 2106  {cab 2715  ∀wral 3064  Vcvv 3432   ⊆ wss 3887   ⊊ wpss 3888  𝒫 cpw 4533  ∪ cuni 4839  ∩ cint 4879  ran crn 5590   ↾ cres 5591  suc csuc 6268  –1-1→wf1 6430  ‘cfv 6433  (class class class)co 7275  ωcom 7712  reccrdg 8240   ↑_m cmap 8615

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241

This theorem is referenced by: (None)