Theorem r1tr 9672

Description: The cumulative hierarchy of sets is transitive. Lemma 7T of [Enderton] p. 202. (Contributed by NM, 8-Sep-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)

Assertion

Ref	Expression
r1tr	⊢ Tr (𝑅1‘𝐴)

Proof of Theorem r1tr

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		r1funlim 9662	. . . . . 6 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
2	1	simpri 485	. . . . 5 ⊢ Lim dom 𝑅1
3		limord 6368	. . . . 5 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
4		ordsson 7719	. . . . 5 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
5	2, 3, 4	mp2b 10	. . . 4 ⊢ dom 𝑅1 ⊆ On
6	5	sseli 3931	. . 3 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ∈ On)
7		fveq2 6822	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = (𝑅1‘∅))
8		r10 9664	. . . . . 6 ⊢ (𝑅1‘∅) = ∅
9	7, 8	eqtrdi 2780	. . . . 5 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = ∅)
10		treq 5206	. . . . 5 ⊢ ((𝑅1‘𝑥) = ∅ → (Tr (𝑅1‘𝑥) ↔ Tr ∅))
11	9, 10	syl 17	. . . 4 ⊢ (𝑥 = ∅ → (Tr (𝑅1‘𝑥) ↔ Tr ∅))
12		fveq2 6822	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑅1‘𝑥) = (𝑅1‘𝑦))
13		treq 5206	. . . . 5 ⊢ ((𝑅1‘𝑥) = (𝑅1‘𝑦) → (Tr (𝑅1‘𝑥) ↔ Tr (𝑅1‘𝑦)))
14	12, 13	syl 17	. . . 4 ⊢ (𝑥 = 𝑦 → (Tr (𝑅1‘𝑥) ↔ Tr (𝑅1‘𝑦)))
15		fveq2 6822	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑦))
16		treq 5206	. . . . 5 ⊢ ((𝑅1‘𝑥) = (𝑅1‘suc 𝑦) → (Tr (𝑅1‘𝑥) ↔ Tr (𝑅1‘suc 𝑦)))
17	15, 16	syl 17	. . . 4 ⊢ (𝑥 = suc 𝑦 → (Tr (𝑅1‘𝑥) ↔ Tr (𝑅1‘suc 𝑦)))
18		fveq2 6822	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑅1‘𝑥) = (𝑅1‘𝐴))
19		treq 5206	. . . . 5 ⊢ ((𝑅1‘𝑥) = (𝑅1‘𝐴) → (Tr (𝑅1‘𝑥) ↔ Tr (𝑅1‘𝐴)))
20	18, 19	syl 17	. . . 4 ⊢ (𝑥 = 𝐴 → (Tr (𝑅1‘𝑥) ↔ Tr (𝑅1‘𝐴)))
21		tr0 5211	. . . 4 ⊢ Tr ∅
22		limsuc 7782	. . . . . . . 8 ⊢ (Lim dom 𝑅1 → (𝑦 ∈ dom 𝑅1 ↔ suc 𝑦 ∈ dom 𝑅1))
23	2, 22	ax-mp 5	. . . . . . 7 ⊢ (𝑦 ∈ dom 𝑅1 ↔ suc 𝑦 ∈ dom 𝑅1)
24		simpr 484	. . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ Tr (𝑅1‘𝑦)) → Tr (𝑅1‘𝑦))
25		pwtr 5395	. . . . . . . . 9 ⊢ (Tr (𝑅1‘𝑦) ↔ Tr 𝒫 (𝑅1‘𝑦))
26	24, 25	sylib 218	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ Tr (𝑅1‘𝑦)) → Tr 𝒫 (𝑅1‘𝑦))
27		r1sucg 9665	. . . . . . . . 9 ⊢ (𝑦 ∈ dom 𝑅1 → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1‘𝑦))
28		treq 5206	. . . . . . . . 9 ⊢ ((𝑅1‘suc 𝑦) = 𝒫 (𝑅1‘𝑦) → (Tr (𝑅1‘suc 𝑦) ↔ Tr 𝒫 (𝑅1‘𝑦)))
29	27, 28	syl 17	. . . . . . . 8 ⊢ (𝑦 ∈ dom 𝑅1 → (Tr (𝑅1‘suc 𝑦) ↔ Tr 𝒫 (𝑅1‘𝑦)))
30	26, 29	syl5ibrcom 247	. . . . . . 7 ⊢ ((𝑦 ∈ On ∧ Tr (𝑅1‘𝑦)) → (𝑦 ∈ dom 𝑅1 → Tr (𝑅1‘suc 𝑦)))
31	23, 30	biimtrrid 243	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ Tr (𝑅1‘𝑦)) → (suc 𝑦 ∈ dom 𝑅1 → Tr (𝑅1‘suc 𝑦)))
32		ndmfv 6855	. . . . . . . 8 ⊢ (¬ suc 𝑦 ∈ dom 𝑅1 → (𝑅1‘suc 𝑦) = ∅)
33		treq 5206	. . . . . . . 8 ⊢ ((𝑅1‘suc 𝑦) = ∅ → (Tr (𝑅1‘suc 𝑦) ↔ Tr ∅))
34	32, 33	syl 17	. . . . . . 7 ⊢ (¬ suc 𝑦 ∈ dom 𝑅1 → (Tr (𝑅1‘suc 𝑦) ↔ Tr ∅))
35	21, 34	mpbiri 258	. . . . . 6 ⊢ (¬ suc 𝑦 ∈ dom 𝑅1 → Tr (𝑅1‘suc 𝑦))
36	31, 35	pm2.61d1 180	. . . . 5 ⊢ ((𝑦 ∈ On ∧ Tr (𝑅1‘𝑦)) → Tr (𝑅1‘suc 𝑦))
37	36	ex 412	. . . 4 ⊢ (𝑦 ∈ On → (Tr (𝑅1‘𝑦) → Tr (𝑅1‘suc 𝑦)))
38		triun 5213	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 Tr (𝑅1‘𝑦) → Tr ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦))
39		r1limg 9667	. . . . . . . . . 10 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ Lim 𝑥) → (𝑅1‘𝑥) = ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦))
40	39	ancoms 458	. . . . . . . . 9 ⊢ ((Lim 𝑥 ∧ 𝑥 ∈ dom 𝑅1) → (𝑅1‘𝑥) = ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦))
41		treq 5206	. . . . . . . . 9 ⊢ ((𝑅1‘𝑥) = ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦) → (Tr (𝑅1‘𝑥) ↔ Tr ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦)))
42	40, 41	syl 17	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ 𝑥 ∈ dom 𝑅1) → (Tr (𝑅1‘𝑥) ↔ Tr ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦)))
43	38, 42	imbitrrid 246	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ 𝑥 ∈ dom 𝑅1) → (∀𝑦 ∈ 𝑥 Tr (𝑅1‘𝑦) → Tr (𝑅1‘𝑥)))
44	43	impancom 451	. . . . . 6 ⊢ ((Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr (𝑅1‘𝑦)) → (𝑥 ∈ dom 𝑅1 → Tr (𝑅1‘𝑥)))
45		ndmfv 6855	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ dom 𝑅1 → (𝑅1‘𝑥) = ∅)
46	45, 10	syl 17	. . . . . . 7 ⊢ (¬ 𝑥 ∈ dom 𝑅1 → (Tr (𝑅1‘𝑥) ↔ Tr ∅))
47	21, 46	mpbiri 258	. . . . . 6 ⊢ (¬ 𝑥 ∈ dom 𝑅1 → Tr (𝑅1‘𝑥))
48	44, 47	pm2.61d1 180	. . . . 5 ⊢ ((Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr (𝑅1‘𝑦)) → Tr (𝑅1‘𝑥))
49	48	ex 412	. . . 4 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 Tr (𝑅1‘𝑦) → Tr (𝑅1‘𝑥)))
50	11, 14, 17, 20, 21, 37, 49	tfinds 7793	. . 3 ⊢ (𝐴 ∈ On → Tr (𝑅1‘𝐴))
51	6, 50	syl 17	. 2 ⊢ (𝐴 ∈ dom 𝑅1 → Tr (𝑅1‘𝐴))
52		ndmfv 6855	. . . 4 ⊢ (¬ 𝐴 ∈ dom 𝑅1 → (𝑅1‘𝐴) = ∅)
53		treq 5206	. . . 4 ⊢ ((𝑅1‘𝐴) = ∅ → (Tr (𝑅1‘𝐴) ↔ Tr ∅))
54	52, 53	syl 17	. . 3 ⊢ (¬ 𝐴 ∈ dom 𝑅1 → (Tr (𝑅1‘𝐴) ↔ Tr ∅))
55	21, 54	mpbiri 258	. 2 ⊢ (¬ 𝐴 ∈ dom 𝑅1 → Tr (𝑅1‘𝐴))
56	51, 55	pm2.61i 182	1 ⊢ Tr (𝑅1‘𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3903  ∅c0 4284  𝒫 cpw 4551  ∪ ciun 4941  Tr wtr 5199  dom cdm 5619  Ord word 6306  Oncon0 6307  Lim wlim 6308  suc csuc 6309  Fun wfun 6476  ‘cfv 6482  𝑅₁cr1 9658

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 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

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-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-r1 9660

This theorem is referenced by:  r1tr2  9673  r1ordg  9674  r1ord3g  9675  r1ord2  9677  r1sssuc  9679  r1pwss  9680  r1val1  9682  rankwflemb  9689  r1elwf  9692  r1elssi  9701  uniwf  9715  tcrank  9780  ackbij2lem3  10134  r1limwun  10630  tskr1om2  10662  inagrud  44289