Theorem r1tr 9695

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 9685	. . . . . 6 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
2	1	simpri 485	. . . . 5 ⊢ Lim dom 𝑅1
3		limord 6380	. . . . 5 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
4		ordsson 7732	. . . . 5 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
5	2, 3, 4	mp2b 10	. . . 4 ⊢ dom 𝑅1 ⊆ On
6	5	sseli 3918	. . 3 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ∈ On)
7		fveq2 6836	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = (𝑅1‘∅))
8		r10 9687	. . . . . 6 ⊢ (𝑅1‘∅) = ∅
9	7, 8	eqtrdi 2788	. . . . 5 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = ∅)
10		treq 5200	. . . . 5 ⊢ ((𝑅1‘𝑥) = ∅ → (Tr (𝑅1‘𝑥) ↔ Tr ∅))
11	9, 10	syl 17	. . . 4 ⊢ (𝑥 = ∅ → (Tr (𝑅1‘𝑥) ↔ Tr ∅))
12		fveq2 6836	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑅1‘𝑥) = (𝑅1‘𝑦))
13		treq 5200	. . . . 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 6836	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑦))
16		treq 5200	. . . . 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 6836	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑅1‘𝑥) = (𝑅1‘𝐴))
19		treq 5200	. . . . 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 5206	. . . 4 ⊢ Tr ∅
22		limsuc 7795	. . . . . . . 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 5401	. . . . . . . . 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 9688	. . . . . . . . 9 ⊢ (𝑦 ∈ dom 𝑅1 → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1‘𝑦))
28		treq 5200	. . . . . . . . 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 6868	. . . . . . . 8 ⊢ (¬ suc 𝑦 ∈ dom 𝑅1 → (𝑅1‘suc 𝑦) = ∅)
33		treq 5200	. . . . . . . 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 5208	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 Tr (𝑅1‘𝑦) → Tr ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦))
39		r1limg 9690	. . . . . . . . . 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 5200	. . . . . . . . 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 6868	. . . . . . . 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 7806	. . 3 ⊢ (𝐴 ∈ On → Tr (𝑅1‘𝐴))
51	6, 50	syl 17	. 2 ⊢ (𝐴 ∈ dom 𝑅1 → Tr (𝑅1‘𝐴))
52		ndmfv 6868	. . . 4 ⊢ (¬ 𝐴 ∈ dom 𝑅1 → (𝑅1‘𝐴) = ∅)
53		treq 5200	. . . 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 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3890  ∅c0 4274  𝒫 cpw 4542  ∪ ciun 4934  Tr wtr 5193  dom cdm 5626  Ord word 6318  Oncon0 6319  Lim wlim 6320  suc csuc 6321  Fun wfun 6488  ‘cfv 6494  𝑅₁cr1 9681

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 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684

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-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7365  df-om 7813  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-r1 9683

This theorem is referenced by:  r1tr2  9696  r1ordg  9697  r1ord3g  9698  r1ord2  9700  r1sssuc  9702  r1pwss  9703  r1val1  9705  rankwflemb  9712  r1elwf  9715  r1elssi  9724  uniwf  9738  tcrank  9803  ackbij2lem3  10157  r1limwun  10654  tskr1om2  10686  ttcwf  36726  inagrud  44747