Theorem oicl 9415

Description: The order type of the well-order 𝑅 on 𝐴 is an ordinal. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)

Hypothesis

Ref	Expression
oicl.1	⊢ 𝐹 = OrdIso(𝑅, 𝐴)

Assertion

Ref	Expression
oicl	⊢ Ord dom 𝐹

Proof of Theorem oicl

Dummy variables 𝑢 𝑡 𝑣 𝑥 ℎ 𝑗 𝑤 𝑧 𝑓 𝑖 𝑟 𝑠 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2731	. . . . 5 ⊢ recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) = recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)))
2		eqid 2731	. . . . 5 ⊢ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}
3		eqid 2731	. . . . 5 ⊢ (ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)) = (ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))
4	1, 2, 3	ordtypecbv 9403	. . . 4 ⊢ recs((𝑓 ∈ V ↦ (℩𝑠 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))) = recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)))
5		eqid 2731	. . . 4 ⊢ {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (recs((𝑓 ∈ V ↦ (℩𝑠 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))) “ 𝑥)𝑧𝑅𝑡} = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (recs((𝑓 ∈ V ↦ (℩𝑠 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))) “ 𝑥)𝑧𝑅𝑡}
6		oicl.1	. . . 4 ⊢ 𝐹 = OrdIso(𝑅, 𝐴)
7		simpl 482	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 We 𝐴)
8		simpr 484	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Se 𝐴)
9	4, 2, 3, 5, 6, 7, 8	ordtypelem5 9408	. . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → (Ord dom 𝐹 ∧ 𝐹:dom 𝐹⟶𝐴))
10	9	simpld 494	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → Ord dom 𝐹)
11		ord0 6360	. . 3 ⊢ Ord ∅
12	6	oi0 9414	. . . . . 6 ⊢ (¬ (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 = ∅)
13	12	dmeqd 5844	. . . . 5 ⊢ (¬ (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → dom 𝐹 = dom ∅)
14		dm0 5859	. . . . 5 ⊢ dom ∅ = ∅
15	13, 14	eqtrdi 2782	. . . 4 ⊢ (¬ (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → dom 𝐹 = ∅)
16		ordeq 6313	. . . 4 ⊢ (dom 𝐹 = ∅ → (Ord dom 𝐹 ↔ Ord ∅))
17	15, 16	syl 17	. . 3 ⊢ (¬ (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → (Ord dom 𝐹 ↔ Ord ∅))
18	11, 17	mpbiri 258	. 2 ⊢ (¬ (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → Ord dom 𝐹)
19	10, 18	pm2.61i 182	1 ⊢ Ord dom 𝐹

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1541  ∀wral 3047  ∃wrex 3056  {crab 3395  Vcvv 3436  ∅c0 4280   class class class wbr 5089   ↦ cmpt 5170   Se wse 5565   We wwe 5566  dom cdm 5614  ran crn 5615   “ cima 5617  Ord word 6305  Oncon0 6306  ⟶wf 6477  ℩crio 7302  recscrecs 8290  OrdIsocoi 9395

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-oi 9396

This theorem is referenced by:  oion  9422  oieu  9425  oismo  9426  oiid  9427  wofib  9431  cantnflt  9562  cantnfp1lem3  9570  cantnflem1b  9576  cantnflem1  9579  wemapwe  9587  cnfcomlem  9589  cnfcom  9590  cnfcom2lem  9591  infxpenlem  9904  hsmexlem1  10317  fpwwe2lem7  10528  fpwwe2lem8  10529  fpwwe2lem9  10530