Theorem fin23lem36 10256

Description: Lemma for fin23 10297. 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 6832	. . . . . . 7 ⊢ (𝑎 = 𝐵 → (𝑌‘𝑎) = (𝑌‘𝐵))
2	1	rneqd 5885	. . . . . 6 ⊢ (𝑎 = 𝐵 → ran (𝑌‘𝑎) = ran (𝑌‘𝐵))
3	2	unieqd 4874	. . . . 5 ⊢ (𝑎 = 𝐵 → ∪ ran (𝑌‘𝑎) = ∪ ran (𝑌‘𝐵))
4	3	sseq1d 3963	. . . 4 ⊢ (𝑎 = 𝐵 → (∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵) ↔ ∪ ran (𝑌‘𝐵) ⊆ ∪ ran (𝑌‘𝐵)))
5	4	imbi2d 340	. . 3 ⊢ (𝑎 = 𝐵 → ((𝜑 → ∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵)) ↔ (𝜑 → ∪ ran (𝑌‘𝐵) ⊆ ∪ ran (𝑌‘𝐵))))
6		fveq2 6832	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑌‘𝑎) = (𝑌‘𝑏))
7	6	rneqd 5885	. . . . . 6 ⊢ (𝑎 = 𝑏 → ran (𝑌‘𝑎) = ran (𝑌‘𝑏))
8	7	unieqd 4874	. . . . 5 ⊢ (𝑎 = 𝑏 → ∪ ran (𝑌‘𝑎) = ∪ ran (𝑌‘𝑏))
9	8	sseq1d 3963	. . . 4 ⊢ (𝑎 = 𝑏 → (∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵) ↔ ∪ ran (𝑌‘𝑏) ⊆ ∪ ran (𝑌‘𝐵)))
10	9	imbi2d 340	. . 3 ⊢ (𝑎 = 𝑏 → ((𝜑 → ∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵)) ↔ (𝜑 → ∪ ran (𝑌‘𝑏) ⊆ ∪ ran (𝑌‘𝐵))))
11		fveq2 6832	. . . . . . 7 ⊢ (𝑎 = suc 𝑏 → (𝑌‘𝑎) = (𝑌‘suc 𝑏))
12	11	rneqd 5885	. . . . . 6 ⊢ (𝑎 = suc 𝑏 → ran (𝑌‘𝑎) = ran (𝑌‘suc 𝑏))
13	12	unieqd 4874	. . . . 5 ⊢ (𝑎 = suc 𝑏 → ∪ ran (𝑌‘𝑎) = ∪ ran (𝑌‘suc 𝑏))
14	13	sseq1d 3963	. . . 4 ⊢ (𝑎 = suc 𝑏 → (∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵) ↔ ∪ ran (𝑌‘suc 𝑏) ⊆ ∪ ran (𝑌‘𝐵)))
15	14	imbi2d 340	. . 3 ⊢ (𝑎 = suc 𝑏 → ((𝜑 → ∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵)) ↔ (𝜑 → ∪ ran (𝑌‘suc 𝑏) ⊆ ∪ ran (𝑌‘𝐵))))
16		fveq2 6832	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑌‘𝑎) = (𝑌‘𝐴))
17	16	rneqd 5885	. . . . . 6 ⊢ (𝑎 = 𝐴 → ran (𝑌‘𝑎) = ran (𝑌‘𝐴))
18	17	unieqd 4874	. . . . 5 ⊢ (𝑎 = 𝐴 → ∪ ran (𝑌‘𝑎) = ∪ ran (𝑌‘𝐴))
19	18	sseq1d 3963	. . . 4 ⊢ (𝑎 = 𝐴 → (∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵) ↔ ∪ ran (𝑌‘𝐴) ⊆ ∪ ran (𝑌‘𝐵)))
20	19	imbi2d 340	. . 3 ⊢ (𝑎 = 𝐴 → ((𝜑 → ∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵)) ↔ (𝜑 → ∪ ran (𝑌‘𝐴) ⊆ ∪ ran (𝑌‘𝐵))))
21		ssid 3954	. . . 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 10255	. . . . . . . 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 4050	. . . . . 6 ⊢ (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵 ⊆ 𝑏 ∧ 𝜑)) → ∪ ran (𝑌‘suc 𝑏) ⊆ ∪ ran (𝑌‘𝑏))
33		sstr2 3938	. . . . . 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 456	. . . 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 7837	. 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝐴) → (𝜑 → ∪ ran (𝑌‘𝐴) ⊆ ∪ ran (𝑌‘𝐵)))
38	37	impr 454	1 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵 ⊆ 𝐴 ∧ 𝜑)) → ∪ ran (𝑌‘𝐴) ⊆ ∪ ran (𝑌‘𝐵))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1539   = wceq 1541   ∈ wcel 2113  {cab 2712  ∀wral 3049  Vcvv 3438   ⊆ wss 3899   ⊊ wpss 3900  𝒫 cpw 4552  ∪ cuni 4861  ∩ cint 4900  ran crn 5623   ↾ cres 5624  suc csuc 6317  –1-1→wf1 6487  ‘cfv 6490  (class class class)co 7356  ωcom 7806  reccrdg 8338   ↑_m cmap 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339

This theorem is referenced by: (None)