Theorem rankwflemb 9778

Description: Two ways of saying a set is well-founded. (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)

Assertion

Ref	Expression
rankwflemb	⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))

Distinct variable group:   𝑥,𝐴

Proof of Theorem rankwflemb

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni 4873	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ (𝑅1 “ On)))
2		eleq2 2851	. . . . . . . 8 ⊢ ((𝑅1‘𝑥) = 𝑦 → (𝐴 ∈ (𝑅1‘𝑥) ↔ 𝐴 ∈ 𝑦))
3	2	biimprcd 253	. . . . . . 7 ⊢ (𝐴 ∈ 𝑦 → ((𝑅1‘𝑥) = 𝑦 → 𝐴 ∈ (𝑅1‘𝑥)))
4		r1tr 9761	. . . . . . . . . . 11 ⊢ Tr (𝑅1‘𝑥)
5		trss 5226	. . . . . . . . . . 11 ⊢ (Tr (𝑅1‘𝑥) → (𝐴 ∈ (𝑅1‘𝑥) → 𝐴 ⊆ (𝑅1‘𝑥)))
6	4, 5	ax-mp 5	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝑅1‘𝑥) → 𝐴 ⊆ (𝑅1‘𝑥))
7		elpwg 4563	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝑅1‘𝑥) → (𝐴 ∈ 𝒫 (𝑅1‘𝑥) ↔ 𝐴 ⊆ (𝑅1‘𝑥)))
8	6, 7	mpbird 260	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑅1‘𝑥) → 𝐴 ∈ 𝒫 (𝑅1‘𝑥))
9		elfvdm 6916	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝑅1‘𝑥) → 𝑥 ∈ dom 𝑅1)
10		r1sucg 9754	. . . . . . . . . 10 ⊢ (𝑥 ∈ dom 𝑅1 → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
11	9, 10	syl 18	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑅1‘𝑥) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
12	8, 11	eleqtrrd 2865	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑅1‘𝑥) → 𝐴 ∈ (𝑅1‘suc 𝑥))
13	12	a1i 11	. . . . . . 7 ⊢ (𝑥 ∈ On → (𝐴 ∈ (𝑅1‘𝑥) → 𝐴 ∈ (𝑅1‘suc 𝑥)))
14	3, 13	syl9 78	. . . . . 6 ⊢ (𝐴 ∈ 𝑦 → (𝑥 ∈ On → ((𝑅1‘𝑥) = 𝑦 → 𝐴 ∈ (𝑅1‘suc 𝑥))))
15	14	reximdvai 3175	. . . . 5 ⊢ (𝐴 ∈ 𝑦 → (∃𝑥 ∈ On (𝑅1‘𝑥) = 𝑦 → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥)))
16		r1funlim 9751	. . . . . . 7 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
17	16	simpli 489	. . . . . 6 ⊢ Fun 𝑅1
18		fvelima 6947	. . . . . 6 ⊢ ((Fun 𝑅1 ∧ 𝑦 ∈ (𝑅1 “ On)) → ∃𝑥 ∈ On (𝑅1‘𝑥) = 𝑦)
19	17, 18	mpan 703	. . . . 5 ⊢ (𝑦 ∈ (𝑅1 “ On) → ∃𝑥 ∈ On (𝑅1‘𝑥) = 𝑦)
20	15, 19	impel 515	. . . 4 ⊢ ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ (𝑅1 “ On)) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
21	20	exlimiv 1963	. . 3 ⊢ (∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ (𝑅1 “ On)) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
22	1, 21	sylbi 220	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
23		elfvdm 6916	. . . . . 6 ⊢ (𝐴 ∈ (𝑅1‘suc 𝑥) → suc 𝑥 ∈ dom 𝑅1)
24		fvelrn 7072	. . . . . 6 ⊢ ((Fun 𝑅1 ∧ suc 𝑥 ∈ dom 𝑅1) → (𝑅1‘suc 𝑥) ∈ ran 𝑅1)
25	17, 23, 24	sylancr 599	. . . . 5 ⊢ (𝐴 ∈ (𝑅1‘suc 𝑥) → (𝑅1‘suc 𝑥) ∈ ran 𝑅1)
26		df-ima 5672	. . . . . 6 ⊢ (𝑅1 “ On) = ran (𝑅1 ↾ On)
27		funrel 6554	. . . . . . . . 9 ⊢ (Fun 𝑅1 → Rel 𝑅1)
28	17, 27	ax-mp 5	. . . . . . . 8 ⊢ Rel 𝑅1
29	16	simpri 491	. . . . . . . . 9 ⊢ Lim dom 𝑅1
30		limord 6423	. . . . . . . . 9 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
31		ordsson 7785	. . . . . . . . 9 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
32	29, 30, 31	mp2b 10	. . . . . . . 8 ⊢ dom 𝑅1 ⊆ On
33		relssres 6019	. . . . . . . 8 ⊢ ((Rel 𝑅1 ∧ dom 𝑅1 ⊆ On) → (𝑅1 ↾ On) = 𝑅1)
34	28, 32, 33	mp2an 705	. . . . . . 7 ⊢ (𝑅1 ↾ On) = 𝑅1
35	34	rneqi 5925	. . . . . 6 ⊢ ran (𝑅1 ↾ On) = ran 𝑅1
36	26, 35	eqtri 2785	. . . . 5 ⊢ (𝑅1 “ On) = ran 𝑅1
37	25, 36	eleqtrrdi 2873	. . . 4 ⊢ (𝐴 ∈ (𝑅1‘suc 𝑥) → (𝑅1‘suc 𝑥) ∈ (𝑅1 “ On))
38		elunii 4875	. . . 4 ⊢ ((𝐴 ∈ (𝑅1‘suc 𝑥) ∧ (𝑅1‘suc 𝑥) ∈ (𝑅1 “ On)) → 𝐴 ∈ ∪ (𝑅1 “ On))
39	37, 38	mpdan 700	. . 3 ⊢ (𝐴 ∈ (𝑅1‘suc 𝑥) → 𝐴 ∈ ∪ (𝑅1 “ On))
40	39	rexlimivw 3161	. 2 ⊢ (∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥) → 𝐴 ∈ ∪ (𝑅1 “ On))
41	22, 40	impbii 212	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃wrex 3088   ⊆ wss 3902  𝒫 cpw 4560  ∪ cuni 4870  Tr wtr 5216  dom cdm 5659  ran crn 5660   ↾ cres 5661   “ cima 5662  Rel wrel 5664  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:  rankf  9779  r1elwf  9781  rankvalb  9782  rankidb  9785  rankwflem  9800  tcrank  9869  dfac12r  10152