Theorem fin23lem36 10332

Description: Lemma for fin23 10373. 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 6882	. . . . . . 7 ⊢ (𝑎 = 𝐵 → (𝑌‘𝑎) = (𝑌‘𝐵))
2	1	rneqd 5929	. . . . . 6 ⊢ (𝑎 = 𝐵 → ran (𝑌‘𝑎) = ran (𝑌‘𝐵))
3	2	unieqd 4887	. . . . 5 ⊢ (𝑎 = 𝐵 → ∪ ran (𝑌‘𝑎) = ∪ ran (𝑌‘𝐵))
4	3	sseq1d 3974	. . . 4 ⊢ (𝑎 = 𝐵 → (∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵) ↔ ∪ ran (𝑌‘𝐵) ⊆ ∪ ran (𝑌‘𝐵)))
5	4	imbi2d 343	. . 3 ⊢ (𝑎 = 𝐵 → ((𝜑 → ∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵)) ↔ (𝜑 → ∪ ran (𝑌‘𝐵) ⊆ ∪ ran (𝑌‘𝐵))))
6		fveq2 6882	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑌‘𝑎) = (𝑌‘𝑏))
7	6	rneqd 5929	. . . . . 6 ⊢ (𝑎 = 𝑏 → ran (𝑌‘𝑎) = ran (𝑌‘𝑏))
8	7	unieqd 4887	. . . . 5 ⊢ (𝑎 = 𝑏 → ∪ ran (𝑌‘𝑎) = ∪ ran (𝑌‘𝑏))
9	8	sseq1d 3974	. . . 4 ⊢ (𝑎 = 𝑏 → (∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵) ↔ ∪ ran (𝑌‘𝑏) ⊆ ∪ ran (𝑌‘𝐵)))
10	9	imbi2d 343	. . 3 ⊢ (𝑎 = 𝑏 → ((𝜑 → ∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵)) ↔ (𝜑 → ∪ ran (𝑌‘𝑏) ⊆ ∪ ran (𝑌‘𝐵))))
11		fveq2 6882	. . . . . . 7 ⊢ (𝑎 = suc 𝑏 → (𝑌‘𝑎) = (𝑌‘suc 𝑏))
12	11	rneqd 5929	. . . . . 6 ⊢ (𝑎 = suc 𝑏 → ran (𝑌‘𝑎) = ran (𝑌‘suc 𝑏))
13	12	unieqd 4887	. . . . 5 ⊢ (𝑎 = suc 𝑏 → ∪ ran (𝑌‘𝑎) = ∪ ran (𝑌‘suc 𝑏))
14	13	sseq1d 3974	. . . 4 ⊢ (𝑎 = suc 𝑏 → (∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵) ↔ ∪ ran (𝑌‘suc 𝑏) ⊆ ∪ ran (𝑌‘𝐵)))
15	14	imbi2d 343	. . 3 ⊢ (𝑎 = suc 𝑏 → ((𝜑 → ∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵)) ↔ (𝜑 → ∪ ran (𝑌‘suc 𝑏) ⊆ ∪ ran (𝑌‘𝐵))))
16		fveq2 6882	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑌‘𝑎) = (𝑌‘𝐴))
17	16	rneqd 5929	. . . . . 6 ⊢ (𝑎 = 𝐴 → ran (𝑌‘𝑎) = ran (𝑌‘𝐴))
18	17	unieqd 4887	. . . . 5 ⊢ (𝑎 = 𝐴 → ∪ ran (𝑌‘𝑎) = ∪ ran (𝑌‘𝐴))
19	18	sseq1d 3974	. . . 4 ⊢ (𝑎 = 𝐴 → (∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵) ↔ ∪ ran (𝑌‘𝐴) ⊆ ∪ ran (𝑌‘𝐵)))
20	19	imbi2d 343	. . 3 ⊢ (𝑎 = 𝐴 → ((𝜑 → ∪ ran (𝑌‘𝑎) ⊆ ∪ ran (𝑌‘𝐵)) ↔ (𝜑 → ∪ ran (𝑌‘𝐴) ⊆ ∪ ran (𝑌‘𝐵))))
21		ssid 3965	. . . 4 ⊢ ∪ ran (𝑌‘𝐵) ⊆ ∪ ran (𝑌‘𝐵)
22	21	2a1i 12	. . 3 ⊢ (𝐵 ∈ ω → (𝜑 → ∪ ran (𝑌‘𝐵) ⊆ ∪ ran (𝑌‘𝐵)))
23		simprr 784	. . . . . . . 8 ⊢ (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵 ⊆ 𝑏 ∧ 𝜑)) → 𝜑)
24		simpll 778	. . . . . . . 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 10331	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ ω) → ∪ ran (𝑌‘suc 𝑏) ⊊ ∪ ran (𝑌‘𝑏))
31	23, 24, 30	syl2anc 595	. . . . . . 7 ⊢ (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵 ⊆ 𝑏 ∧ 𝜑)) → ∪ ran (𝑌‘suc 𝑏) ⊊ ∪ ran (𝑌‘𝑏))
32	31	pssssd 4060	. . . . . 6 ⊢ (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵 ⊆ 𝑏 ∧ 𝜑)) → ∪ ran (𝑌‘suc 𝑏) ⊆ ∪ ran (𝑌‘𝑏))
33		sstr2 3950	. . . . . 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 18	. . . . 5 ⊢ (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵 ⊆ 𝑏 ∧ 𝜑)) → (∪ ran (𝑌‘𝑏) ⊆ ∪ ran (𝑌‘𝐵) → ∪ ran (𝑌‘suc 𝑏) ⊆ ∪ ran (𝑌‘𝐵)))
35	34	expr 461	. . . 4 ⊢ (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑏) → (𝜑 → (∪ ran (𝑌‘𝑏) ⊆ ∪ ran (𝑌‘𝐵) → ∪ ran (𝑌‘suc 𝑏) ⊆ ∪ ran (𝑌‘𝐵))))
36	35	a2d 30	. . 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 7894	. 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝐴) → (𝜑 → ∪ ran (𝑌‘𝐴) ⊆ ∪ ran (𝑌‘𝐵)))
38	37	impr 459	1 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵 ⊆ 𝐴 ∧ 𝜑)) → ∪ ran (𝑌‘𝐴) ⊆ ∪ ran (𝑌‘𝐵))

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1565   = wceq 1567   ∈ wcel 2149  {cab 2747  ∀wral 3085  Vcvv 3461   ⊆ wss 3911   ⊊ wpss 3912  𝒫 cpw 4565  ∪ cuni 4874  ∩ cint 4914  ran crn 5663   ↾ cres 5664  suc csuc 6363  –1-1→wf1 6534  ‘cfv 6537  (class class class)co 7411  ωcom 7862  reccrdg 8396   ↑_m cmap 8824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5259  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-om 7863  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397

This theorem is referenced by: (None)