Theorem r1tr 4634

Description: The cumulative hierarchy of sets is transitive. Lemma 7T of [Enderton] p. 202.

Assertion

Ref	Expression
r1tr	|- Tr (R1` A)

Proof of Theorem r1tr

Step	Hyp	Ref	Expression
1		fveq2 3715	. . . 4 |- (x = (/) -> (R1` x) = (R1` (/)))
2		treq 2681	. . . 4 |- ((R1` x) = (R1` (/)) -> (Tr (R1` x) <-> Tr (R1` (/))))
3	1, 2	syl 10	. . 3 |- (x = (/) -> (Tr (R1` x) <-> Tr (R1` (/))))
4		fveq2 3715	. . . 4 |- (x = y -> (R1` x) = (R1` y))
5		treq 2681	. . . 4 |- ((R1` x) = (R1` y) -> (Tr (R1` x) <-> Tr (R1` y)))
6	4, 5	syl 10	. . 3 |- (x = y -> (Tr (R1` x) <-> Tr (R1` y)))
7		fveq2 3715	. . . 4 |- (x = suc y -> (R1` x) = (R1` suc y))
8		treq 2681	. . . 4 |- ((R1` x) = (R1` suc y) -> (Tr (R1` x) <-> Tr (R1` suc y)))
9	7, 8	syl 10	. . 3 |- (x = suc y -> (Tr (R1` x) <-> Tr (R1` suc y)))
10		fveq2 3715	. . . 4 |- (x = A -> (R1` x) = (R1` A))
11		treq 2681	. . . 4 |- ((R1` x) = (R1` A) -> (Tr (R1` x) <-> Tr (R1` A)))
12	10, 11	syl 10	. . 3 |- (x = A -> (Tr (R1` x) <-> Tr (R1` A)))
13		tr0 2686	. . . 4 |- Tr (/)
14		r10 4631	. . . . 5 |- (R1` (/)) = (/)
15		treq 2681	. . . . 5 |- ((R1` (/)) = (/) -> (Tr (R1` (/)) <-> Tr (/)))
16	14, 15	ax-mp 7	. . . 4 |- (Tr (R1` (/)) <-> Tr (/))
17	13, 16	mpbir 190	. . 3 |- Tr (R1` (/))
18		r1suc 4632	. . . . . . . . . 10 |- (y e. On -> (R1` suc y) = P~(R1` y))
19	18	eleq2d 1538	. . . . . . . . 9 |- (y e. On -> (x e. (R1` suc y) <-> x e. P~(R1` y)))
20		visset 1809	. . . . . . . . . 10 |- x e. V
21	20	elpw 2400	. . . . . . . . 9 |- (x e. P~(R1` y) <-> x (_ (R1` y))
22	19, 21	syl6bb 535	. . . . . . . 8 |- (y e. On -> (x e. (R1` suc y) <-> x (_ (R1` y)))
23	22	adantr 389	. . . . . . 7 |- ((y e. On /\ Tr (R1` y)) -> (x e. (R1` suc y) <-> x (_ (R1` y)))
24		ssel 2059	. . . . . . . . . 10 |- (x (_ (R1` y) -> (z e. x -> z e. (R1` y)))
25		dftr4 2680	. . . . . . . . . . . 12 |- (Tr (R1` y) <-> (R1` y) (_ P~(R1` y))
26		ssel 2059	. . . . . . . . . . . 12 |- ((R1` y) (_ P~(R1` y) -> (z e. (R1` y) -> z e. P~(R1` y)))
27	25, 26	sylbi 199	. . . . . . . . . . 11 |- (Tr (R1` y) -> (z e. (R1` y) -> z e. P~(R1` y)))
28	18	eleq2d 1538	. . . . . . . . . . . 12 |- (y e. On -> (z e. (R1` suc y) <-> z e. P~(R1` y)))
29	28	biimprd 154	. . . . . . . . . . 11 |- (y e. On -> (z e. P~(R1` y) -> z e. (R1` suc y)))
30	27, 29	sylan9r 469	. . . . . . . . . 10 |- ((y e. On /\ Tr (R1` y)) -> (z e. (R1` y) -> z e. (R1` suc y)))
31	24, 30	sylan9r 469	. . . . . . . . 9 |- (((y e. On /\ Tr (R1` y)) /\ x (_ (R1` y)) -> (z e. x -> z e. (R1` suc y)))
32	31	ssrdv 2066	. . . . . . . 8 |- (((y e. On /\ Tr (R1` y)) /\ x (_ (R1` y)) -> x (_ (R1` suc y))
33	32	ex 373	. . . . . . 7 |- ((y e. On /\ Tr (R1` y)) -> (x (_ (R1` y) -> x (_ (R1` suc y)))
34	23, 33	sylbid 203	. . . . . 6 |- ((y e. On /\ Tr (R1` y)) -> (x e. (R1` suc y) -> x (_ (R1` suc y)))
35	34	r19.21aiv 1710	. . . . 5 |- ((y e. On /\ Tr (R1` y)) -> A.x e. (R1` suc y)x (_ (R1` suc y))
36		dftr3 2679	. . . . 5 |- (Tr (R1` suc y) <-> A.x e. (R1` suc y)x (_ (R1` suc y))
37	35, 36	sylibr 200	. . . 4 |- ((y e. On /\ Tr (R1` y)) -> Tr (R1` suc y))
38	37	ex 373	. . 3 |- (y e. On -> (Tr (R1` y) -> Tr (R1` suc y)))
39		r1lim 4633	. . . . . . . . . . 11 |- ((x e. V /\ Lim x) -> (R1` x) = U_y e. x (R1` y))
40	20, 39	mpan 694	. . . . . . . . . 10 |- (Lim x -> (R1` x) = U_y e. x (R1` y))
41	40	eleq2d 1538	. . . . . . . . 9 |- (Lim x -> (z e. (R1` x) <-> z e. U_y e. x (R1` y)))
42		eliun 2565	. . . . . . . . . 10 |- (z e. U_y e. x (R1` y) <-> E.y e. x z e. (R1` y))
43	42	biimp 151	. . . . . . . . 9 |- (z e. U_y e. x (R1` y) -> E.y e. x z e. (R1` y))
44	41, 43	syl6bi 214	. . . . . . . 8 |- (Lim x -> (z e. (R1` x) -> E.y e. x z e. (R1` y)))
45		hbra1 1684	. . . . . . . . 9 |- (A.y e. x Tr (R1` y) -> A.yA.y e. x Tr (R1` y))
46		ra4 1691	. . . . . . . . . 10 |- (A.y e. x Tr (R1` y) -> (y e. x -> Tr (R1` y)))
47		trss 2684	. . . . . . . . . 10 |- (Tr (R1` y) -> (z e. (R1` y) -> z (_ (R1` y)))
48	46, 47	syl6 22	. . . . . . . . 9 |- (A.y e. x Tr (R1` y) -> (y e. x -> (z e. (R1` y) -> z (_ (R1` y))))
49	45, 48	r19.22d 1732	. . . . . . . 8 |- (A.y e. x Tr (R1` y) -> (E.y e. x z e. (R1` y) -> E.y e. x z (_ (R1` y)))
50	44, 49	sylan9 468	. . . . . . 7 |- ((Lim x /\ A.y e. x Tr (R1` y)) -> (z e. (R1` x) -> E.y e. x z (_ (R1` y)))
51	40	sseq2d 2085	. . . . . . . . 9 |- (Lim x -> (z (_ (R1` x) <-> z (_ U_y e. x (R1` y)))
52		ssiun 2587	. . . . . . . . 9 |- (E.y e. x z (_ (R1` y) -> z (_ U_y e. x (R1` y))
53	51, 52	syl5bir 210	. . . . . . . 8 |- (Lim x -> (E.y e. x z (_ (R1` y) -> z (_ (R1` x)))
54	53	adantr 389	. . . . . . 7 |- ((Lim x /\ A.y e. x Tr (R1` y)) -> (E.y e. x z (_ (R1` y) -> z (_ (R1` x)))
55	50, 54	syld 27	. . . . . 6 |- ((Lim x /\ A.y e. x Tr (R1` y)) -> (z e. (R1` x) -> z (_ (R1` x)))
56	55	r19.21aiv 1710	. . . . 5 |- ((Lim x /\ A.y e. x Tr (R1` y)) -> A.z e. (R1` x)z (_ (R1` x))
57		dftr3 2679	. . . . 5 |- (Tr (R1` x) <-> A.z e. (R1` x)z (_ (R1` x))
58	56, 57	sylibr 200	. . . 4 |- ((Lim x /\ A.y e. x Tr (R1` y)) -> Tr (R1` x))
59	58	ex 373	. . 3 |- (Lim x -> (A.y e. x Tr (R1` y) -> Tr (R1` x)))
60	3, 6, 9, 12, 17, 38, 59	tfinds 3156	. 2 |- (A e. On -> Tr (R1` A))
61		r1fnon 4630	. . . . . . . 8 |- R1 Fn On
62		fndm 3579	. . . . . . . 8 |- (R1 Fn On -> dom R1 = On)
63	61, 62	ax-mp 7	. . . . . . 7 |- dom R1 = On
64	63	eleq2i 1535	. . . . . 6 |- (A e. dom R1 <-> A e. On)
65	64	negbii 187	. . . . 5 |- (-. A e. dom R1 <-> -. A e. On)
66		ndmfv 3736	. . . . 5 |- (-. A e. dom R1 -> (R1` A) = (/))
67	65, 66	sylbir 201	. . . 4 |- (-. A e. On -> (R1` A) = (/))
68		treq 2681	. . . 4 |- ((R1` A) = (/) -> (Tr (R1` A) <-> Tr (/)))
69	67, 68	syl 10	. . 3 |- (-. A e. On -> (Tr (R1` A) <-> Tr (/)))
70	13, 69	mpbiri 194	. 2 |- (-. A e. On -> Tr (R1` A))
71	60, 70	pm2.61i 126	1 |- Tr (R1` A)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  A.wral 1642  E.wrex 1643  Vcvv 1807   (_ wss 2043  (/)c0 2276  P~cpw 2397  U_ciun 2561  Tr wtr 2675  Oncon0 2943  Lim wlim 2944  suc csuc 2945  dom cdm 3165   Fn wfn 3172  ` cfv 3177  R1cr1 4621

This theorem is referenced by:  r1ord 4635  r1ord2 4636

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193  df-rdg 3923  df-r1 4623

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