Theorem r1tr 9761

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 9751	. . . . . 6 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
2	1	simpri 491	. . . . 5 ⊢ Lim dom 𝑅1
3		limord 6423	. . . . 5 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
4		ordsson 7785	. . . . 5 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
5	2, 3, 4	mp2b 10	. . . 4 ⊢ dom 𝑅1 ⊆ On
6	5	sseli 3930	. . 3 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ∈ On)
7		fveq2 6882	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = (𝑅1‘∅))
8		r10 9753	. . . . . 6 ⊢ (𝑅1‘∅) = ∅
9	7, 8	eqtrdi 2813	. . . . 5 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = ∅)
10		treq 5223	. . . . 5 ⊢ ((𝑅1‘𝑥) = ∅ → (Tr (𝑅1‘𝑥) ↔ Tr ∅))
11	9, 10	syl 18	. . . 4 ⊢ (𝑥 = ∅ → (Tr (𝑅1‘𝑥) ↔ Tr ∅))
12		fveq2 6882	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑅1‘𝑥) = (𝑅1‘𝑦))
13		treq 5223	. . . . 5 ⊢ ((𝑅1‘𝑥) = (𝑅1‘𝑦) → (Tr (𝑅1‘𝑥) ↔ Tr (𝑅1‘𝑦)))
14	12, 13	syl 18	. . . 4 ⊢ (𝑥 = 𝑦 → (Tr (𝑅1‘𝑥) ↔ Tr (𝑅1‘𝑦)))
15		fveq2 6882	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑦))
16		treq 5223	. . . . 5 ⊢ ((𝑅1‘𝑥) = (𝑅1‘suc 𝑦) → (Tr (𝑅1‘𝑥) ↔ Tr (𝑅1‘suc 𝑦)))
17	15, 16	syl 18	. . . 4 ⊢ (𝑥 = suc 𝑦 → (Tr (𝑅1‘𝑥) ↔ Tr (𝑅1‘suc 𝑦)))
18		fveq2 6882	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑅1‘𝑥) = (𝑅1‘𝐴))
19		treq 5223	. . . . 5 ⊢ ((𝑅1‘𝑥) = (𝑅1‘𝐴) → (Tr (𝑅1‘𝑥) ↔ Tr (𝑅1‘𝐴)))
20	18, 19	syl 18	. . . 4 ⊢ (𝑥 = 𝐴 → (Tr (𝑅1‘𝑥) ↔ Tr (𝑅1‘𝐴)))
21		tr0 5229	. . . 4 ⊢ Tr ∅
22		limsuc 7848	. . . . . . . 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		pwtr 5431	. . . . . . . . 9 ⊢ (Tr (𝑅1‘𝑦) ↔ Tr 𝒫 (𝑅1‘𝑦))
25	24	bilani 510	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ Tr (𝑅1‘𝑦)) → Tr 𝒫 (𝑅1‘𝑦))
26		r1sucg 9754	. . . . . . . . 9 ⊢ (𝑦 ∈ dom 𝑅1 → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1‘𝑦))
27		treq 5223	. . . . . . . . 9 ⊢ ((𝑅1‘suc 𝑦) = 𝒫 (𝑅1‘𝑦) → (Tr (𝑅1‘suc 𝑦) ↔ Tr 𝒫 (𝑅1‘𝑦)))
28	26, 27	syl 18	. . . . . . . 8 ⊢ (𝑦 ∈ dom 𝑅1 → (Tr (𝑅1‘suc 𝑦) ↔ Tr 𝒫 (𝑅1‘𝑦)))
29	25, 28	syl5ibrcom 250	. . . . . . 7 ⊢ ((𝑦 ∈ On ∧ Tr (𝑅1‘𝑦)) → (𝑦 ∈ dom 𝑅1 → Tr (𝑅1‘suc 𝑦)))
30	23, 29	biimtrrid 246	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ Tr (𝑅1‘𝑦)) → (suc 𝑦 ∈ dom 𝑅1 → Tr (𝑅1‘suc 𝑦)))
31		ndmfv 6914	. . . . . . . 8 ⊢ (¬ suc 𝑦 ∈ dom 𝑅1 → (𝑅1‘suc 𝑦) = ∅)
32		treq 5223	. . . . . . . 8 ⊢ ((𝑅1‘suc 𝑦) = ∅ → (Tr (𝑅1‘suc 𝑦) ↔ Tr ∅))
33	31, 32	syl 18	. . . . . . 7 ⊢ (¬ suc 𝑦 ∈ dom 𝑅1 → (Tr (𝑅1‘suc 𝑦) ↔ Tr ∅))
34	21, 33	mpbiri 261	. . . . . 6 ⊢ (¬ suc 𝑦 ∈ dom 𝑅1 → Tr (𝑅1‘suc 𝑦))
35	30, 34	pm2.61d1 182	. . . . 5 ⊢ ((𝑦 ∈ On ∧ Tr (𝑅1‘𝑦)) → Tr (𝑅1‘suc 𝑦))
36	35	ex 418	. . . 4 ⊢ (𝑦 ∈ On → (Tr (𝑅1‘𝑦) → Tr (𝑅1‘suc 𝑦)))
37		triun 5231	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 Tr (𝑅1‘𝑦) → Tr ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦))
38		r1limg 9756	. . . . . . . . . 10 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ Lim 𝑥) → (𝑅1‘𝑥) = ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦))
39	38	ancoms 464	. . . . . . . . 9 ⊢ ((Lim 𝑥 ∧ 𝑥 ∈ dom 𝑅1) → (𝑅1‘𝑥) = ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦))
40		treq 5223	. . . . . . . . 9 ⊢ ((𝑅1‘𝑥) = ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦) → (Tr (𝑅1‘𝑥) ↔ Tr ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦)))
41	39, 40	syl 18	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ 𝑥 ∈ dom 𝑅1) → (Tr (𝑅1‘𝑥) ↔ Tr ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦)))
42	37, 41	imbitrrid 249	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ 𝑥 ∈ dom 𝑅1) → (∀𝑦 ∈ 𝑥 Tr (𝑅1‘𝑦) → Tr (𝑅1‘𝑥)))
43	42	impancom 457	. . . . . 6 ⊢ ((Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr (𝑅1‘𝑦)) → (𝑥 ∈ dom 𝑅1 → Tr (𝑅1‘𝑥)))
44		ndmfv 6914	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ dom 𝑅1 → (𝑅1‘𝑥) = ∅)
45	44, 10	syl 18	. . . . . . 7 ⊢ (¬ 𝑥 ∈ dom 𝑅1 → (Tr (𝑅1‘𝑥) ↔ Tr ∅))
46	21, 45	mpbiri 261	. . . . . 6 ⊢ (¬ 𝑥 ∈ dom 𝑅1 → Tr (𝑅1‘𝑥))
47	43, 46	pm2.61d1 182	. . . . 5 ⊢ ((Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr (𝑅1‘𝑦)) → Tr (𝑅1‘𝑥))
48	47	ex 418	. . . 4 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 Tr (𝑅1‘𝑦) → Tr (𝑅1‘𝑥)))
49	11, 14, 17, 20, 21, 36, 48	tfinds 7859	. . 3 ⊢ (𝐴 ∈ On → Tr (𝑅1‘𝐴))
50	6, 49	syl 18	. 2 ⊢ (𝐴 ∈ dom 𝑅1 → Tr (𝑅1‘𝐴))
51		ndmfv 6914	. . . 4 ⊢ (¬ 𝐴 ∈ dom 𝑅1 → (𝑅1‘𝐴) = ∅)
52		treq 5223	. . . 4 ⊢ ((𝑅1‘𝐴) = ∅ → (Tr (𝑅1‘𝐴) ↔ Tr ∅))
53	51, 52	syl 18	. . 3 ⊢ (¬ 𝐴 ∈ dom 𝑅1 → (Tr (𝑅1‘𝐴) ↔ Tr ∅))
54	21, 53	mpbiri 261	. 2 ⊢ (¬ 𝐴 ∈ dom 𝑅1 → Tr (𝑅1‘𝐴))
55	50, 54	pm2.61i 184	1 ⊢ Tr (𝑅1‘𝐴)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078   ⊆ wss 3902  ∅c0 4282  𝒫 cpw 4560  ∪ ciun 4954  Tr wtr 5216  dom cdm 5659  Ord word 6360  Oncon0 6361  Lim wlim 6362  suc csuc 6363  Fun wfun 6531  ‘cfv 6537  𝑅₁cr1 9747

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-om 7866  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-r1 9749

This theorem is used by:  r1tr2  9762  r1ordg  9763  r1ord3g  9764  r1ord2  9766  r1sssuc  9768  r1pwss  9769  r1val1  9771  rankwflemb  9778  r1elwf  9781  r1elssi  9790  uniwf  9804  tcrank  9869  ackbij2lem3  10245  r1limwun  10748  tskr1om2  10780  ttcwf  37130  inagrud  45107