Theorem r1tr 9755

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 9745	. . . . . 6 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
2	1	simpri 486	. . . . 5 ⊢ Lim dom 𝑅1
3		limord 6414	. . . . 5 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
4		ordsson 7754	. . . . 5 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
5	2, 3, 4	mp2b 10	. . . 4 ⊢ dom 𝑅1 ⊆ On
6	5	sseli 3975	. . 3 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ∈ On)
7		fveq2 6879	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = (𝑅1‘∅))
8		r10 9747	. . . . . 6 ⊢ (𝑅1‘∅) = ∅
9	7, 8	eqtrdi 2788	. . . . 5 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = ∅)
10		treq 5267	. . . . 5 ⊢ ((𝑅1‘𝑥) = ∅ → (Tr (𝑅1‘𝑥) ↔ Tr ∅))
11	9, 10	syl 17	. . . 4 ⊢ (𝑥 = ∅ → (Tr (𝑅1‘𝑥) ↔ Tr ∅))
12		fveq2 6879	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑅1‘𝑥) = (𝑅1‘𝑦))
13		treq 5267	. . . . 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 6879	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑦))
16		treq 5267	. . . . 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 6879	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑅1‘𝑥) = (𝑅1‘𝐴))
19		treq 5267	. . . . 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 5272	. . . 4 ⊢ Tr ∅
22		limsuc 7822	. . . . . . . 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 485	. . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ Tr (𝑅1‘𝑦)) → Tr (𝑅1‘𝑦))
25		pwtr 5446	. . . . . . . . 9 ⊢ (Tr (𝑅1‘𝑦) ↔ Tr 𝒫 (𝑅1‘𝑦))
26	24, 25	sylib 217	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ Tr (𝑅1‘𝑦)) → Tr 𝒫 (𝑅1‘𝑦))
27		r1sucg 9748	. . . . . . . . 9 ⊢ (𝑦 ∈ dom 𝑅1 → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1‘𝑦))
28		treq 5267	. . . . . . . . 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 246	. . . . . . 7 ⊢ ((𝑦 ∈ On ∧ Tr (𝑅1‘𝑦)) → (𝑦 ∈ dom 𝑅1 → Tr (𝑅1‘suc 𝑦)))
31	23, 30	biimtrrid 242	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ Tr (𝑅1‘𝑦)) → (suc 𝑦 ∈ dom 𝑅1 → Tr (𝑅1‘suc 𝑦)))
32		ndmfv 6914	. . . . . . . 8 ⊢ (¬ suc 𝑦 ∈ dom 𝑅1 → (𝑅1‘suc 𝑦) = ∅)
33		treq 5267	. . . . . . . 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 257	. . . . . 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 413	. . . 4 ⊢ (𝑦 ∈ On → (Tr (𝑅1‘𝑦) → Tr (𝑅1‘suc 𝑦)))
38		triun 5274	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 Tr (𝑅1‘𝑦) → Tr ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦))
39		r1limg 9750	. . . . . . . . . 10 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ Lim 𝑥) → (𝑅1‘𝑥) = ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦))
40	39	ancoms 459	. . . . . . . . 9 ⊢ ((Lim 𝑥 ∧ 𝑥 ∈ dom 𝑅1) → (𝑅1‘𝑥) = ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦))
41		treq 5267	. . . . . . . . 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 245	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ 𝑥 ∈ dom 𝑅1) → (∀𝑦 ∈ 𝑥 Tr (𝑅1‘𝑦) → Tr (𝑅1‘𝑥)))
44	43	impancom 452	. . . . . 6 ⊢ ((Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr (𝑅1‘𝑦)) → (𝑥 ∈ dom 𝑅1 → Tr (𝑅1‘𝑥)))
45		ndmfv 6914	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ dom 𝑅1 → (𝑅1‘𝑥) = ∅)
46	45, 10	syl 17	. . . . . . 7 ⊢ (¬ 𝑥 ∈ dom 𝑅1 → (Tr (𝑅1‘𝑥) ↔ Tr ∅))
47	21, 46	mpbiri 257	. . . . . 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 413	. . . 4 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 Tr (𝑅1‘𝑦) → Tr (𝑅1‘𝑥)))
50	11, 14, 17, 20, 21, 37, 49	tfinds 7833	. . 3 ⊢ (𝐴 ∈ On → Tr (𝑅1‘𝐴))
51	6, 50	syl 17	. 2 ⊢ (𝐴 ∈ dom 𝑅1 → Tr (𝑅1‘𝐴))
52		ndmfv 6914	. . . 4 ⊢ (¬ 𝐴 ∈ dom 𝑅1 → (𝑅1‘𝐴) = ∅)
53		treq 5267	. . . 4 ⊢ ((𝑅1‘𝐴) = ∅ → (Tr (𝑅1‘𝐴) ↔ Tr ∅))
54	52, 53	syl 17	. . 3 ⊢ (¬ 𝐴 ∈ dom 𝑅1 → (Tr (𝑅1‘𝐴) ↔ Tr ∅))
55	21, 54	mpbiri 257	. 2 ⊢ (¬ 𝐴 ∈ dom 𝑅1 → Tr (𝑅1‘𝐴))
56	51, 55	pm2.61i 182	1 ⊢ Tr (𝑅1‘𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061   ⊆ wss 3945  ∅c0 4319  𝒫 cpw 4597  ∪ ciun 4991  Tr wtr 5259  dom cdm 5670  Ord word 6353  Oncon0 6354  Lim wlim 6355  suc csuc 6356  Fun wfun 6527  ‘cfv 6533  𝑅₁cr1 9741

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7397  df-om 7840  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8355  df-rdg 8394  df-r1 9743

This theorem is referenced by:  r1tr2  9756  r1ordg  9757  r1ord3g  9758  r1ord2  9760  r1sssuc  9762  r1pwss  9763  r1val1  9765  rankwflemb  9772  r1elwf  9775  r1elssi  9784  uniwf  9798  tcrank  9863  ackbij2lem3  10220  r1limwun  10715  tskr1om2  10747  inagrud  42890