Theorem ordtypelem8 9442

Description: Lemma for ordtype 9449. (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 9438	. . . . 5 ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
9	8	fdmd 6680	. . . 4 ⊢ (𝜑 → dom 𝑂 = (𝑇 ∩ dom 𝐹))
10		inss1 4191	. . . . 5 ⊢ (𝑇 ∩ dom 𝐹) ⊆ 𝑇
11	1, 2, 3, 4, 5, 6, 7	ordtypelem2 9436	. . . . . 6 ⊢ (𝜑 → Ord 𝑇)
12		ordsson 7738	. . . . . 6 ⊢ (Ord 𝑇 → 𝑇 ⊆ On)
13	11, 12	syl 17	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ On)
14	10, 13	sstrid 3947	. . . 4 ⊢ (𝜑 → (𝑇 ∩ dom 𝐹) ⊆ On)
15	9, 14	eqsstrd 3970	. . 3 ⊢ (𝜑 → dom 𝑂 ⊆ On)
16		epweon 7730	. . . 4 ⊢ E We On
17		weso 5623	. . . 4 ⊢ ( E We On → E Or On)
18	16, 17	ax-mp 5	. . 3 ⊢ E Or On
19		soss 5560	. . 3 ⊢ (dom 𝑂 ⊆ On → ( E Or On → E Or dom 𝑂))
20	15, 18, 19	mpisyl 21	. 2 ⊢ (𝜑 → E Or dom 𝑂)
21	8	frnd 6678	. . . 4 ⊢ (𝜑 → ran 𝑂 ⊆ 𝐴)
22		wess 5618	. . . 4 ⊢ (ran 𝑂 ⊆ 𝐴 → (𝑅 We 𝐴 → 𝑅 We ran 𝑂))
23	21, 6, 22	sylc 65	. . 3 ⊢ (𝜑 → 𝑅 We ran 𝑂)
24		weso 5623	. . 3 ⊢ (𝑅 We ran 𝑂 → 𝑅 Or ran 𝑂)
25		sopo 5559	. . 3 ⊢ (𝑅 Or ran 𝑂 → 𝑅 Po ran 𝑂)
26	23, 24, 25	3syl 18	. 2 ⊢ (𝜑 → 𝑅 Po ran 𝑂)
27	8	ffund 6674	. . 3 ⊢ (𝜑 → Fun 𝑂)
28		funforn 6761	. . 3 ⊢ (Fun 𝑂 ↔ 𝑂:dom 𝑂–onto→ran 𝑂)
29	27, 28	sylib 218	. 2 ⊢ (𝜑 → 𝑂:dom 𝑂–onto→ran 𝑂)
30		epel 5535	. . . . 5 ⊢ (𝑎 E 𝑏 ↔ 𝑎 ∈ 𝑏)
31	1, 2, 3, 4, 5, 6, 7	ordtypelem6 9440	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ dom 𝑂) → (𝑎 ∈ 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏)))
32	30, 31	biimtrid 242	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ dom 𝑂) → (𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏)))
33	32	ralrimiva 3130	. . 3 ⊢ (𝜑 → ∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏)))
34	33	ralrimivw 3134	. 2 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑂∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏)))
35		soisoi 7284	. 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 839	1 ⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  {crab 3401  Vcvv 3442   ∩ cin 3902   ⊆ wss 3903   class class class wbr 5100   ↦ cmpt 5181   E cep 5531   Po wpo 5538   Or wor 5539   Se wse 5583   We wwe 5584  dom cdm 5632  ran crn 5633   “ cima 5635  Ord word 6324  Oncon0 6325  Fun wfun 6494  –onto→wfo 6498  ‘cfv 6500   Isom wiso 6501  ℩crio 7324  recscrecs 8312  OrdIsocoi 9426

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-oi 9427

This theorem is referenced by:  ordtypelem9  9443  ordtypelem10  9444  oiiso2  9448