Theorem fin23lem35 10385

Description: Lemma for fin23 10427. Strict 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
fin23lem35	⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ∪ ran (𝑌‘suc 𝐴) ⊊ ∪ ran (𝑌‘𝐴))

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

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

Proof of Theorem fin23lem35

Step	Hyp	Ref	Expression
1		fin23lem33.f	. . . . 5 ⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
2		fin23lem.f	. . . . 5 ⊢ (𝜑 → ℎ:ω–1-1→V)
3		fin23lem.g	. . . . 5 ⊢ (𝜑 → ∪ ran ℎ ⊆ 𝐺)
4		fin23lem.h	. . . . 5 ⊢ (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) → ((𝑖‘𝑗):ω–1-1→V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗)))
5		fin23lem.i	. . . . 5 ⊢ 𝑌 = (rec(𝑖, ℎ) ↾ ω)
6	1, 2, 3, 4, 5	fin23lem34 10384	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ((𝑌‘𝐴):ω–1-1→V ∧ ∪ ran (𝑌‘𝐴) ⊆ 𝐺))
7		fvex 6920	. . . . . . 7 ⊢ (𝑌‘𝐴) ∈ V
8		f1eq1 6800	. . . . . . . . 9 ⊢ (𝑗 = (𝑌‘𝐴) → (𝑗:ω–1-1→V ↔ (𝑌‘𝐴):ω–1-1→V))
9		rneq 5950	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑌‘𝐴) → ran 𝑗 = ran (𝑌‘𝐴))
10	9	unieqd 4925	. . . . . . . . . 10 ⊢ (𝑗 = (𝑌‘𝐴) → ∪ ran 𝑗 = ∪ ran (𝑌‘𝐴))
11	10	sseq1d 4027	. . . . . . . . 9 ⊢ (𝑗 = (𝑌‘𝐴) → (∪ ran 𝑗 ⊆ 𝐺 ↔ ∪ ran (𝑌‘𝐴) ⊆ 𝐺))
12	8, 11	anbi12d 632	. . . . . . . 8 ⊢ (𝑗 = (𝑌‘𝐴) → ((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) ↔ ((𝑌‘𝐴):ω–1-1→V ∧ ∪ ran (𝑌‘𝐴) ⊆ 𝐺)))
13		fveq2 6907	. . . . . . . . . 10 ⊢ (𝑗 = (𝑌‘𝐴) → (𝑖‘𝑗) = (𝑖‘(𝑌‘𝐴)))
14		f1eq1 6800	. . . . . . . . . 10 ⊢ ((𝑖‘𝑗) = (𝑖‘(𝑌‘𝐴)) → ((𝑖‘𝑗):ω–1-1→V ↔ (𝑖‘(𝑌‘𝐴)):ω–1-1→V))
15	13, 14	syl 17	. . . . . . . . 9 ⊢ (𝑗 = (𝑌‘𝐴) → ((𝑖‘𝑗):ω–1-1→V ↔ (𝑖‘(𝑌‘𝐴)):ω–1-1→V))
16	13	rneqd 5952	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑌‘𝐴) → ran (𝑖‘𝑗) = ran (𝑖‘(𝑌‘𝐴)))
17	16	unieqd 4925	. . . . . . . . . 10 ⊢ (𝑗 = (𝑌‘𝐴) → ∪ ran (𝑖‘𝑗) = ∪ ran (𝑖‘(𝑌‘𝐴)))
18	17, 10	psseq12d 4107	. . . . . . . . 9 ⊢ (𝑗 = (𝑌‘𝐴) → (∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗 ↔ ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴)))
19	15, 18	anbi12d 632	. . . . . . . 8 ⊢ (𝑗 = (𝑌‘𝐴) → (((𝑖‘𝑗):ω–1-1→V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗) ↔ ((𝑖‘(𝑌‘𝐴)):ω–1-1→V ∧ ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴))))
20	12, 19	imbi12d 344	. . . . . . 7 ⊢ (𝑗 = (𝑌‘𝐴) → (((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) → ((𝑖‘𝑗):ω–1-1→V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗)) ↔ (((𝑌‘𝐴):ω–1-1→V ∧ ∪ ran (𝑌‘𝐴) ⊆ 𝐺) → ((𝑖‘(𝑌‘𝐴)):ω–1-1→V ∧ ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴)))))
21	7, 20	spcv 3605	. . . . . 6 ⊢ (∀𝑗((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) → ((𝑖‘𝑗):ω–1-1→V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗)) → (((𝑌‘𝐴):ω–1-1→V ∧ ∪ ran (𝑌‘𝐴) ⊆ 𝐺) → ((𝑖‘(𝑌‘𝐴)):ω–1-1→V ∧ ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴))))
22	4, 21	syl 17	. . . . 5 ⊢ (𝜑 → (((𝑌‘𝐴):ω–1-1→V ∧ ∪ ran (𝑌‘𝐴) ⊆ 𝐺) → ((𝑖‘(𝑌‘𝐴)):ω–1-1→V ∧ ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴))))
23	22	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (((𝑌‘𝐴):ω–1-1→V ∧ ∪ ran (𝑌‘𝐴) ⊆ 𝐺) → ((𝑖‘(𝑌‘𝐴)):ω–1-1→V ∧ ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴))))
24	6, 23	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ((𝑖‘(𝑌‘𝐴)):ω–1-1→V ∧ ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴)))
25	24	simprd 495	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴))
26		frsuc 8476	. . . . . . 7 ⊢ (𝐴 ∈ ω → ((rec(𝑖, ℎ) ↾ ω)‘suc 𝐴) = (𝑖‘((rec(𝑖, ℎ) ↾ ω)‘𝐴)))
27	26	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ((rec(𝑖, ℎ) ↾ ω)‘suc 𝐴) = (𝑖‘((rec(𝑖, ℎ) ↾ ω)‘𝐴)))
28	5	fveq1i 6908	. . . . . 6 ⊢ (𝑌‘suc 𝐴) = ((rec(𝑖, ℎ) ↾ ω)‘suc 𝐴)
29	5	fveq1i 6908	. . . . . . 7 ⊢ (𝑌‘𝐴) = ((rec(𝑖, ℎ) ↾ ω)‘𝐴)
30	29	fveq2i 6910	. . . . . 6 ⊢ (𝑖‘(𝑌‘𝐴)) = (𝑖‘((rec(𝑖, ℎ) ↾ ω)‘𝐴))
31	27, 28, 30	3eqtr4g 2800	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝑌‘suc 𝐴) = (𝑖‘(𝑌‘𝐴)))
32	31	rneqd 5952	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ran (𝑌‘suc 𝐴) = ran (𝑖‘(𝑌‘𝐴)))
33	32	unieqd 4925	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ∪ ran (𝑌‘suc 𝐴) = ∪ ran (𝑖‘(𝑌‘𝐴)))
34	33	psseq1d 4105	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (∪ ran (𝑌‘suc 𝐴) ⊊ ∪ ran (𝑌‘𝐴) ↔ ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴)))
35	25, 34	mpbird 257	1 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ∪ ran (𝑌‘suc 𝐴) ⊊ ∪ ran (𝑌‘𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535   = wceq 1537   ∈ wcel 2106  {cab 2712  ∀wral 3059  Vcvv 3478   ⊆ wss 3963   ⊊ wpss 3964  𝒫 cpw 4605  ∪ cuni 4912  ∩ cint 4951  ran crn 5690   ↾ cres 5691  suc csuc 6388  –1-1→wf1 6560  ‘cfv 6563  (class class class)co 7431  ωcom 7887  reccrdg 8448   ↑_m cmap 8865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449

This theorem is referenced by:  fin23lem36  10386  fin23lem38  10387  fin23lem39  10388