Theorem ordtypelem8 9490

Description: Lemma for ordtype 9497. (Contributed by Mario Carneiro, 25-Jun-2015.)

Hypotheses

Ref	Expression
ordtypelem.1	⊢ 𝐹 = recs(𝐺)
ordtypelem.2	⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}
ordtypelem.3	⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣))
ordtypelem.5	⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}
ordtypelem.6	⊢ 𝑂 = OrdIso(𝑅, 𝐴)
ordtypelem.7	⊢ (𝜑 → 𝑅 We 𝐴)
ordtypelem.8	⊢ (𝜑 → 𝑅 Se 𝐴)

Assertion

Ref	Expression
ordtypelem8	⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))

Distinct variable groups:   𝑣,𝑢,𝐶   ℎ,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧,𝑅   𝐴,ℎ,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧   𝑡,𝑂,𝑢,𝑣,𝑥   𝜑,𝑡,𝑥   ℎ,𝐹,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧

Allowed substitution hints:   𝜑(𝑧, 𝑤, 𝑣, 𝑢, ℎ, 𝑗)   𝐶(𝑥, 𝑧, 𝑤, 𝑡, ℎ, 𝑗)   𝑇(𝑥, 𝑧, 𝑤, 𝑣, 𝑢, 𝑡, ℎ, 𝑗)   𝐺(𝑥, 𝑧, 𝑤, 𝑣, 𝑢, 𝑡, ℎ, 𝑗)   𝑂(𝑧, 𝑤, ℎ, 𝑗)

Proof of Theorem ordtypelem8

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordtypelem.1	. . . . . 6 ⊢ 𝐹 = recs(𝐺)
2		ordtypelem.2	. . . . . 6 ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}
3		ordtypelem.3	. . . . . 6 ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣))
4		ordtypelem.5	. . . . . 6 ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}
5		ordtypelem.6	. . . . . 6 ⊢ 𝑂 = OrdIso(𝑅, 𝐴)
6		ordtypelem.7	. . . . . 6 ⊢ (𝜑 → 𝑅 We 𝐴)
7		ordtypelem.8	. . . . . 6 ⊢ (𝜑 → 𝑅 Se 𝐴)
8	1, 2, 3, 4, 5, 6, 7	ordtypelem4 9486	. . . . 5 ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
9	8	fdmd 6720	. . . 4 ⊢ (𝜑 → dom 𝑂 = (𝑇 ∩ dom 𝐹))
10		inss1 4189	. . . . 5 ⊢ (𝑇 ∩ dom 𝐹) ⊆ 𝑇
11	1, 2, 3, 4, 5, 6, 7	ordtypelem2 9484	. . . . . 6 ⊢ (𝜑 → Ord 𝑇)
12		ordsson 7784	. . . . . 6 ⊢ (Ord 𝑇 → 𝑇 ⊆ On)
13	11, 12	syl 18	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ On)
14	10, 13	sstrid 3949	. . . 4 ⊢ (𝜑 → (𝑇 ∩ dom 𝐹) ⊆ On)
15	9, 14	eqsstrd 3972	. . 3 ⊢ (𝜑 → dom 𝑂 ⊆ On)
16		epweon 7776	. . . 4 ⊢ E We On
17		weso 5654	. . . 4 ⊢ ( E We On → E Or On)
18	16, 17	ax-mp 5	. . 3 ⊢ E Or On
19		soss 5591	. . 3 ⊢ (dom 𝑂 ⊆ On → ( E Or On → E Or dom 𝑂))
20	15, 18, 19	mpisyl 22	. 2 ⊢ (𝜑 → E Or dom 𝑂)
21	8	frnd 6718	. . . 4 ⊢ (𝜑 → ran 𝑂 ⊆ 𝐴)
22		wess 5649	. . . 4 ⊢ (ran 𝑂 ⊆ 𝐴 → (𝑅 We 𝐴 → 𝑅 We ran 𝑂))
23	21, 6, 22	sylc 66	. . 3 ⊢ (𝜑 → 𝑅 We ran 𝑂)
24		weso 5654	. . 3 ⊢ (𝑅 We ran 𝑂 → 𝑅 Or ran 𝑂)
25		sopo 5590	. . 3 ⊢ (𝑅 Or ran 𝑂 → 𝑅 Po ran 𝑂)
26	23, 24, 25	3syl 19	. 2 ⊢ (𝜑 → 𝑅 Po ran 𝑂)
27	8	ffund 6714	. . 3 ⊢ (𝜑 → Fun 𝑂)
28		funforn 6803	. . 3 ⊢ (Fun 𝑂 ↔ 𝑂:dom 𝑂–onto→ran 𝑂)
29	27, 28	sylib 221	. 2 ⊢ (𝜑 → 𝑂:dom 𝑂–onto→ran 𝑂)
30		epel 5566	. . . . 5 ⊢ (𝑎 E 𝑏 ↔ 𝑎 ∈ 𝑏)
31	1, 2, 3, 4, 5, 6, 7	ordtypelem6 9488	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ dom 𝑂) → (𝑎 ∈ 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏)))
32	30, 31	biimtrid 245	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ dom 𝑂) → (𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏)))
33	32	ralrimiva 3159	. . 3 ⊢ (𝜑 → ∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏)))
34	33	ralrimivw 3163	. 2 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑂∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏)))
35		soisoi 7332	. 2 ⊢ ((( E Or dom 𝑂 ∧ 𝑅 Po ran 𝑂) ∧ (𝑂:dom 𝑂–onto→ran 𝑂 ∧ ∀𝑎 ∈ dom 𝑂∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏)))) → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
36	20, 26, 29, 34, 35	syl22anc 852	1 ⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  {crab 3418  Vcvv 3457   ∩ cin 3905   ⊆ wss 3906   class class class wbr 5111   ↦ cmpt 5194   E cep 5562   Po wpo 5569   Or wor 5570   Se wse 5614   We wwe 5615  dom cdm 5663  ran crn 5664   “ cima 5666  Ord word 6363  Oncon0 6364  Fun wfun 6534  –onto→wfo 6538  ‘cfv 6540   Isom wiso 6541  ℩crio 7372  recscrecs 8359  OrdIsocoi 9474

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7373  df-ov 7419  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-oi 9475

This theorem is used by:  ordtypelem9  9491  ordtypelem10  9492  oiiso2  9496