Theorem ordtypelem10 9522

Description: Lemma for ordtype 9527. Using ax-rep 5286, 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 9520	. 2 ⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
9	1, 2, 3, 4, 5, 6, 7	ordtypelem4 9516	. . . . 5 ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
10	9	frnd 6726	. . . 4 ⊢ (𝜑 → ran 𝑂 ⊆ 𝐴)
11		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑏 ∈ 𝐴)
12	6	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑅 We 𝐴)
13	7	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑅 Se 𝐴)
14	9	ffund 6722	. . . . . . . . . . . 12 ⊢ (𝜑 → Fun 𝑂)
15	14	funfnd 6580	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑂 Fn dom 𝑂)
16	15	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 Fn dom 𝑂)
17	1, 2, 3, 4, 5, 12, 13	ordtypelem8 9520	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
18		isof1o 7320	. . . . . . . . . . . 12 ⊢ (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) → 𝑂:dom 𝑂–1-1-onto→ran 𝑂)
19		f1of1 6833	. . . . . . . . . . . 12 ⊢ (𝑂:dom 𝑂–1-1-onto→ran 𝑂 → 𝑂:dom 𝑂–1-1→ran 𝑂)
20	17, 18, 19	3syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂:dom 𝑂–1-1→ran 𝑂)
21		simpl 484	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂) → 𝑏 ∈ 𝐴)
22		seex 5639	. . . . . . . . . . . . 13 ⊢ ((𝑅 Se 𝐴 ∧ 𝑏 ∈ 𝐴) → {𝑐 ∈ 𝐴 ∣ 𝑐𝑅𝑏} ∈ V)
23	7, 21, 22	syl2an 597	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → {𝑐 ∈ 𝐴 ∣ 𝑐𝑅𝑏} ∈ V)
24	10	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 ⊆ 𝐴)
25		rexnal 3101	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑚 ∈ dom 𝑂 ¬ (𝑂‘𝑚)𝑅𝑏 ↔ ¬ ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏)
26	1, 2, 3, 4, 5, 6, 7	ordtypelem7 9519	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑚 ∈ dom 𝑂) → ((𝑂‘𝑚)𝑅𝑏 ∨ 𝑏 ∈ ran 𝑂))
27	26	ord 863	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑚 ∈ dom 𝑂) → (¬ (𝑂‘𝑚)𝑅𝑏 → 𝑏 ∈ ran 𝑂))
28	27	rexlimdva 3156	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (∃𝑚 ∈ dom 𝑂 ¬ (𝑂‘𝑚)𝑅𝑏 → 𝑏 ∈ ran 𝑂))
29	25, 28	biimtrrid 242	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (¬ ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏 → 𝑏 ∈ ran 𝑂))
30	29	con1d 145	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (¬ 𝑏 ∈ ran 𝑂 → ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏))
31	30	impr 456	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏)
32		breq1 5152	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = (𝑂‘𝑚) → (𝑐𝑅𝑏 ↔ (𝑂‘𝑚)𝑅𝑏))
33	32	ralrn 7090	. . . . . . . . . . . . . . 15 ⊢ (𝑂 Fn dom 𝑂 → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏))
34	16, 33	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏))
35	31, 34	mpbird 257	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏)
36		ssrab 4071	. . . . . . . . . . . . 13 ⊢ (ran 𝑂 ⊆ {𝑐 ∈ 𝐴 ∣ 𝑐𝑅𝑏} ↔ (ran 𝑂 ⊆ 𝐴 ∧ ∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏))
37	24, 35, 36	sylanbrc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 ⊆ {𝑐 ∈ 𝐴 ∣ 𝑐𝑅𝑏})
38	23, 37	ssexd 5325	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 ∈ V)
39		f1dmex 7943	. . . . . . . . . . 11 ⊢ ((𝑂:dom 𝑂–1-1→ran 𝑂 ∧ ran 𝑂 ∈ V) → dom 𝑂 ∈ V)
40	20, 38, 39	syl2anc 585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → dom 𝑂 ∈ V)
41	16, 40	fnexd 7220	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 ∈ V)
42	1, 2, 3, 4, 5, 12, 13, 41	ordtypelem9 9521	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))
43		isof1o 7320	. . . . . . . 8 ⊢ (𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴) → 𝑂:dom 𝑂–1-1-onto→𝐴)
44		f1ofo 6841	. . . . . . . 8 ⊢ (𝑂:dom 𝑂–1-1-onto→𝐴 → 𝑂:dom 𝑂–onto→𝐴)
45		forn 6809	. . . . . . . 8 ⊢ (𝑂:dom 𝑂–onto→𝐴 → ran 𝑂 = 𝐴)
46	42, 43, 44, 45	4syl 19	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 = 𝐴)
47	11, 46	eleqtrrd 2837	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑏 ∈ ran 𝑂)
48	47	expr 458	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (¬ 𝑏 ∈ ran 𝑂 → 𝑏 ∈ ran 𝑂))
49	48	pm2.18d 127	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → 𝑏 ∈ ran 𝑂)
50	10, 49	eqelssd 4004	. . 3 ⊢ (𝜑 → ran 𝑂 = 𝐴)
51		isoeq5 7318	. . 3 ⊢ (ran 𝑂 = 𝐴 → (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) ↔ 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴)))
52	50, 51	syl 17	. 2 ⊢ (𝜑 → (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) ↔ 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴)))
53	8, 52	mpbid 231	1 ⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071  {crab 3433  Vcvv 3475   ∩ cin 3948   ⊆ wss 3949   class class class wbr 5149   ↦ cmpt 5232   E cep 5580   Se wse 5630   We wwe 5631  dom cdm 5677  ran crn 5678   “ cima 5680  Oncon0 6365   Fn wfn 6539  –1-1→wf1 6541  –onto→wfo 6542  –1-1-onto→wf1o 6543  ‘cfv 6544   Isom wiso 6545  ℩crio 7364  recscrecs 8370  OrdIsocoi 9504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-oi 9505

This theorem is referenced by:  ordtype  9527