Theorem ordtypelem4 9474

Description: Lemma for ordtype 9485. (Contributed by Mario Carneiro, 24-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
ordtypelem4	⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶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 ordtypelem4

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordtypelem.1	. . . . . . . 8 ⊢ 𝐹 = recs(𝐺)
2	1	tfr1a 8362	. . . . . . 7 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹)
3	2	simpli 483	. . . . . 6 ⊢ Fun 𝐹
4		funres 6558	. . . . . 6 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝑇))
5	3, 4	mp1i 13	. . . . 5 ⊢ (𝜑 → Fun (𝐹 ↾ 𝑇))
6	5	funfnd 6547	. . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝑇) Fn dom (𝐹 ↾ 𝑇))
7		dmres 5983	. . . . 5 ⊢ dom (𝐹 ↾ 𝑇) = (𝑇 ∩ dom 𝐹)
8	7	fneq2i 6616	. . . 4 ⊢ ((𝐹 ↾ 𝑇) Fn dom (𝐹 ↾ 𝑇) ↔ (𝐹 ↾ 𝑇) Fn (𝑇 ∩ dom 𝐹))
9	6, 8	sylib 218	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝑇) Fn (𝑇 ∩ dom 𝐹))
10		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → 𝑎 ∈ (𝑇 ∩ dom 𝐹))
11	10	elin1d 4167	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → 𝑎 ∈ 𝑇)
12	11	fvresd 6878	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → ((𝐹 ↾ 𝑇)‘𝑎) = (𝐹‘𝑎))
13		ssrab2 4043	. . . . . . 7 ⊢ {𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣} ⊆ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤}
14		ssrab2 4043	. . . . . . 7 ⊢ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ⊆ 𝐴
15	13, 14	sstri 3956	. . . . . 6 ⊢ {𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣} ⊆ 𝐴
16		ordtypelem.2	. . . . . . 7 ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}
17		ordtypelem.3	. . . . . . 7 ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣))
18		ordtypelem.5	. . . . . . 7 ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}
19		ordtypelem.6	. . . . . . 7 ⊢ 𝑂 = OrdIso(𝑅, 𝐴)
20		ordtypelem.7	. . . . . . 7 ⊢ (𝜑 → 𝑅 We 𝐴)
21		ordtypelem.8	. . . . . . 7 ⊢ (𝜑 → 𝑅 Se 𝐴)
22	1, 16, 17, 18, 19, 20, 21	ordtypelem3 9473	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹‘𝑎) ∈ {𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
23	15, 22	sselid 3944	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹‘𝑎) ∈ 𝐴)
24	12, 23	eqeltrd 2828	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → ((𝐹 ↾ 𝑇)‘𝑎) ∈ 𝐴)
25	24	ralrimiva 3125	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ (𝑇 ∩ dom 𝐹)((𝐹 ↾ 𝑇)‘𝑎) ∈ 𝐴)
26		ffnfv 7091	. . 3 ⊢ ((𝐹 ↾ 𝑇):(𝑇 ∩ dom 𝐹)⟶𝐴 ↔ ((𝐹 ↾ 𝑇) Fn (𝑇 ∩ dom 𝐹) ∧ ∀𝑎 ∈ (𝑇 ∩ dom 𝐹)((𝐹 ↾ 𝑇)‘𝑎) ∈ 𝐴))
27	9, 25, 26	sylanbrc 583	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝑇):(𝑇 ∩ dom 𝐹)⟶𝐴)
28	1, 16, 17, 18, 19, 20, 21	ordtypelem1 9471	. . 3 ⊢ (𝜑 → 𝑂 = (𝐹 ↾ 𝑇))
29	28	feq1d 6670	. 2 ⊢ (𝜑 → (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 ↔ (𝐹 ↾ 𝑇):(𝑇 ∩ dom 𝐹)⟶𝐴))
30	27, 29	mpbird 257	1 ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  {crab 3405  Vcvv 3447   ∩ cin 3913   class class class wbr 5107   ↦ cmpt 5188   Se wse 5589   We wwe 5590  dom cdm 5638  ran crn 5639   ↾ cres 5640   “ cima 5641  Oncon0 6332  Lim wlim 6333  Fun wfun 6505   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  ℩crio 7343  recscrecs 8339  OrdIsocoi 9462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-oi 9463

This theorem is referenced by:  ordtypelem5  9475  ordtypelem6  9476  ordtypelem7  9477  ordtypelem8  9478  ordtypelem9  9479  ordtypelem10  9480