Theorem ordtypelem10 8975

Description: Lemma for ordtype 8980. Using ax-rep 5154, exclude the possibility that 𝑂 is a proper class and does not enumerate all of 𝐴. (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
ordtypelem10	⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))

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 ordtypelem10

Dummy variables 𝑏 𝑐 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordtypelem.1	. . 3 ⊢ 𝐹 = recs(𝐺)
2		ordtypelem.2	. . 3 ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}
3		ordtypelem.3	. . 3 ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣))
4		ordtypelem.5	. . 3 ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}
5		ordtypelem.6	. . 3 ⊢ 𝑂 = OrdIso(𝑅, 𝐴)
6		ordtypelem.7	. . 3 ⊢ (𝜑 → 𝑅 We 𝐴)
7		ordtypelem.8	. . 3 ⊢ (𝜑 → 𝑅 Se 𝐴)
8	1, 2, 3, 4, 5, 6, 7	ordtypelem8 8973	. 2 ⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
9	1, 2, 3, 4, 5, 6, 7	ordtypelem4 8969	. . . . 5 ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
10	9	frnd 6494	. . . 4 ⊢ (𝜑 → ran 𝑂 ⊆ 𝐴)
11		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑏 ∈ 𝐴)
12	6	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑅 We 𝐴)
13	7	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑅 Se 𝐴)
14	1, 2, 3, 4, 5, 12, 13	ordtypelem8 8973	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
15		isof1o 7055	. . . . . . . . . . 11 ⊢ (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) → 𝑂:dom 𝑂–1-1-onto→ran 𝑂)
16		f1of 6590	. . . . . . . . . . 11 ⊢ (𝑂:dom 𝑂–1-1-onto→ran 𝑂 → 𝑂:dom 𝑂⟶ran 𝑂)
17	14, 15, 16	3syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂:dom 𝑂⟶ran 𝑂)
18		f1of1 6589	. . . . . . . . . . . 12 ⊢ (𝑂:dom 𝑂–1-1-onto→ran 𝑂 → 𝑂:dom 𝑂–1-1→ran 𝑂)
19	14, 15, 18	3syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂:dom 𝑂–1-1→ran 𝑂)
20		simpl 486	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂) → 𝑏 ∈ 𝐴)
21		seex 5482	. . . . . . . . . . . . 13 ⊢ ((𝑅 Se 𝐴 ∧ 𝑏 ∈ 𝐴) → {𝑐 ∈ 𝐴 ∣ 𝑐𝑅𝑏} ∈ V)
22	7, 20, 21	syl2an 598	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → {𝑐 ∈ 𝐴 ∣ 𝑐𝑅𝑏} ∈ V)
23	10	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 ⊆ 𝐴)
24		rexnal 3201	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑚 ∈ dom 𝑂 ¬ (𝑂‘𝑚)𝑅𝑏 ↔ ¬ ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏)
25	1, 2, 3, 4, 5, 6, 7	ordtypelem7 8972	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑚 ∈ dom 𝑂) → ((𝑂‘𝑚)𝑅𝑏 ∨ 𝑏 ∈ ran 𝑂))
26	25	ord 861	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑚 ∈ dom 𝑂) → (¬ (𝑂‘𝑚)𝑅𝑏 → 𝑏 ∈ ran 𝑂))
27	26	rexlimdva 3243	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (∃𝑚 ∈ dom 𝑂 ¬ (𝑂‘𝑚)𝑅𝑏 → 𝑏 ∈ ran 𝑂))
28	24, 27	syl5bir 246	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (¬ ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏 → 𝑏 ∈ ran 𝑂))
29	28	con1d 147	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (¬ 𝑏 ∈ ran 𝑂 → ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏))
30	29	impr 458	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏)
31	9	ffund 6491	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → Fun 𝑂)
32	31	funfnd 6355	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑂 Fn dom 𝑂)
33	32	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 Fn dom 𝑂)
34		breq1 5033	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = (𝑂‘𝑚) → (𝑐𝑅𝑏 ↔ (𝑂‘𝑚)𝑅𝑏))
35	34	ralrn 6831	. . . . . . . . . . . . . . 15 ⊢ (𝑂 Fn dom 𝑂 → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏))
36	33, 35	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏))
37	30, 36	mpbird 260	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏)
38		ssrab 4000	. . . . . . . . . . . . 13 ⊢ (ran 𝑂 ⊆ {𝑐 ∈ 𝐴 ∣ 𝑐𝑅𝑏} ↔ (ran 𝑂 ⊆ 𝐴 ∧ ∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏))
39	23, 37, 38	sylanbrc 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 ⊆ {𝑐 ∈ 𝐴 ∣ 𝑐𝑅𝑏})
40	22, 39	ssexd 5192	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 ∈ V)
41		f1dmex 7640	. . . . . . . . . . 11 ⊢ ((𝑂:dom 𝑂–1-1→ran 𝑂 ∧ ran 𝑂 ∈ V) → dom 𝑂 ∈ V)
42	19, 40, 41	syl2anc 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → dom 𝑂 ∈ V)
43		fex 6966	. . . . . . . . . 10 ⊢ ((𝑂:dom 𝑂⟶ran 𝑂 ∧ dom 𝑂 ∈ V) → 𝑂 ∈ V)
44	17, 42, 43	syl2anc 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 ∈ V)
45	1, 2, 3, 4, 5, 12, 13, 44	ordtypelem9 8974	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))
46		isof1o 7055	. . . . . . . 8 ⊢ (𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴) → 𝑂:dom 𝑂–1-1-onto→𝐴)
47		f1ofo 6597	. . . . . . . 8 ⊢ (𝑂:dom 𝑂–1-1-onto→𝐴 → 𝑂:dom 𝑂–onto→𝐴)
48		forn 6568	. . . . . . . 8 ⊢ (𝑂:dom 𝑂–onto→𝐴 → ran 𝑂 = 𝐴)
49	45, 46, 47, 48	4syl 19	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 = 𝐴)
50	11, 49	eleqtrrd 2893	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑏 ∈ ran 𝑂)
51	50	expr 460	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (¬ 𝑏 ∈ ran 𝑂 → 𝑏 ∈ ran 𝑂))
52	51	pm2.18d 127	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → 𝑏 ∈ ran 𝑂)
53	10, 52	eqelssd 3936	. . 3 ⊢ (𝜑 → ran 𝑂 = 𝐴)
54		isoeq5 7053	. . 3 ⊢ (ran 𝑂 = 𝐴 → (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) ↔ 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴)))
55	53, 54	syl 17	. 2 ⊢ (𝜑 → (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) ↔ 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴)))
56	8, 55	mpbid 235	1 ⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  {crab 3110  Vcvv 3441   ∩ cin 3880   ⊆ wss 3881   class class class wbr 5030   ↦ cmpt 5110   E cep 5429   Se wse 5476   We wwe 5477  dom cdm 5519  ran crn 5520   “ cima 5522  Oncon0 6159   Fn wfn 6319  ⟶wf 6320  –1-1→wf1 6321  –onto→wfo 6322  –1-1-onto→wf1o 6323  ‘cfv 6324   Isom wiso 6325  ℩crio 7092  recscrecs 7990  OrdIsocoi 8957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-wrecs 7930  df-recs 7991  df-oi 8958

This theorem is referenced by:  ordtype  8980