Theorem oicl 9477

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 2762	. . . . 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 2762	. . . . 5 ⊢ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}
3		eqid 2762	. . . . 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 9465	. . . 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 2762	. . . 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 486	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 We 𝐴)
8		simpr 488	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Se 𝐴)
9	4, 2, 3, 5, 6, 7, 8	ordtypelem5 9470	. . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → (Ord dom 𝐹 ∧ 𝐹:dom 𝐹⟶𝐴))
10	9	simpld 498	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → Ord dom 𝐹)
11		ord0 6400	. . 3 ⊢ Ord ∅
12	6	oi0 9476	. . . . . 6 ⊢ (¬ (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 = ∅)
13	12	dmeqd 5881	. . . . 5 ⊢ (¬ (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → dom 𝐹 = dom ∅)
14		dm0 5896	. . . . 5 ⊢ dom ∅ = ∅
15	13, 14	eqtrdi 2813	. . . 4 ⊢ (¬ (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → dom 𝐹 = ∅)
16		ordeq 6353	. . . 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 260	. 2 ⊢ (¬ (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → Ord dom 𝐹)
19	10, 18	pm2.61i 183	1 ⊢ Ord dom 𝐹

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 399   = wceq 1560  ∀wral 3076  ∃wrex 3086  {crab 3414  Vcvv 3454  ∅c0 4285   class class class wbr 5100   ↦ cmpt 5181   Se wse 5598   We wwe 5599  dom cdm 5647  ran crn 5648   “ cima 5650  Ord word 6345  Oncon0 6346  ⟶wf 6517  ℩crio 7352  recscrecs 8341  OrdIsocoi 9457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-oi 9458

This theorem is referenced by:  oion  9484  oieu  9487  oismo  9488  oiid  9489  wofib  9493  cantnflt  9627  cantnfp1lem3  9635  cantnflem1b  9641  cantnflem1  9644  wemapwe  9652  cnfcomlem  9654  cnfcom  9655  cnfcom2lem  9656  infxpenlem  9969  hsmexlem1  10383  fpwwe2lem7  10595  fpwwe2lem8  10596  fpwwe2lem9  10597  ordtypeon  35383