Theorem rankwflemb 8904

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 4629	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ (𝑅1 “ On)))
2		r1funlim 8877	. . . . . . . 8 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
3	2	simpli 477	. . . . . . 7 ⊢ Fun 𝑅1
4		fvelima 6471	. . . . . . 7 ⊢ ((Fun 𝑅1 ∧ 𝑦 ∈ (𝑅1 “ On)) → ∃𝑥 ∈ On (𝑅1‘𝑥) = 𝑦)
5	3, 4	mpan 682	. . . . . 6 ⊢ (𝑦 ∈ (𝑅1 “ On) → ∃𝑥 ∈ On (𝑅1‘𝑥) = 𝑦)
6		eleq2 2865	. . . . . . . . 9 ⊢ ((𝑅1‘𝑥) = 𝑦 → (𝐴 ∈ (𝑅1‘𝑥) ↔ 𝐴 ∈ 𝑦))
7	6	biimprcd 242	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑦 → ((𝑅1‘𝑥) = 𝑦 → 𝐴 ∈ (𝑅1‘𝑥)))
8		r1tr 8887	. . . . . . . . . . . 12 ⊢ Tr (𝑅1‘𝑥)
9		trss 4952	. . . . . . . . . . . 12 ⊢ (Tr (𝑅1‘𝑥) → (𝐴 ∈ (𝑅1‘𝑥) → 𝐴 ⊆ (𝑅1‘𝑥)))
10	8, 9	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝑅1‘𝑥) → 𝐴 ⊆ (𝑅1‘𝑥))
11		elpwg 4355	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝑅1‘𝑥) → (𝐴 ∈ 𝒫 (𝑅1‘𝑥) ↔ 𝐴 ⊆ (𝑅1‘𝑥)))
12	10, 11	mpbird 249	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝑅1‘𝑥) → 𝐴 ∈ 𝒫 (𝑅1‘𝑥))
13		elfvdm 6441	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝑅1‘𝑥) → 𝑥 ∈ dom 𝑅1)
14		r1sucg 8880	. . . . . . . . . . 11 ⊢ (𝑥 ∈ dom 𝑅1 → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
15	13, 14	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝑅1‘𝑥) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
16	12, 15	eleqtrrd 2879	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑅1‘𝑥) → 𝐴 ∈ (𝑅1‘suc 𝑥))
17	16	a1i 11	. . . . . . . 8 ⊢ (𝑥 ∈ On → (𝐴 ∈ (𝑅1‘𝑥) → 𝐴 ∈ (𝑅1‘suc 𝑥)))
18	7, 17	syl9 77	. . . . . . 7 ⊢ (𝐴 ∈ 𝑦 → (𝑥 ∈ On → ((𝑅1‘𝑥) = 𝑦 → 𝐴 ∈ (𝑅1‘suc 𝑥))))
19	18	reximdvai 3193	. . . . . 6 ⊢ (𝐴 ∈ 𝑦 → (∃𝑥 ∈ On (𝑅1‘𝑥) = 𝑦 → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥)))
20	5, 19	syl5 34	. . . . 5 ⊢ (𝐴 ∈ 𝑦 → (𝑦 ∈ (𝑅1 “ On) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥)))
21	20	imp 396	. . . 4 ⊢ ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ (𝑅1 “ On)) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
22	21	exlimiv 2026	. . 3 ⊢ (∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ (𝑅1 “ On)) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
23	1, 22	sylbi 209	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
24		elfvdm 6441	. . . . . 6 ⊢ (𝐴 ∈ (𝑅1‘suc 𝑥) → suc 𝑥 ∈ dom 𝑅1)
25		fvelrn 6576	. . . . . 6 ⊢ ((Fun 𝑅1 ∧ suc 𝑥 ∈ dom 𝑅1) → (𝑅1‘suc 𝑥) ∈ ran 𝑅1)
26	3, 24, 25	sylancr 582	. . . . 5 ⊢ (𝐴 ∈ (𝑅1‘suc 𝑥) → (𝑅1‘suc 𝑥) ∈ ran 𝑅1)
27		df-ima 5323	. . . . . 6 ⊢ (𝑅1 “ On) = ran (𝑅1 ↾ On)
28		funrel 6116	. . . . . . . . 9 ⊢ (Fun 𝑅1 → Rel 𝑅1)
29	3, 28	ax-mp 5	. . . . . . . 8 ⊢ Rel 𝑅1
30	2	simpri 480	. . . . . . . . 9 ⊢ Lim dom 𝑅1
31		limord 5998	. . . . . . . . 9 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
32		ordsson 7221	. . . . . . . . 9 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
33	30, 31, 32	mp2b 10	. . . . . . . 8 ⊢ dom 𝑅1 ⊆ On
34		relssres 5646	. . . . . . . 8 ⊢ ((Rel 𝑅1 ∧ dom 𝑅1 ⊆ On) → (𝑅1 ↾ On) = 𝑅1)
35	29, 33, 34	mp2an 684	. . . . . . 7 ⊢ (𝑅1 ↾ On) = 𝑅1
36	35	rneqi 5553	. . . . . 6 ⊢ ran (𝑅1 ↾ On) = ran 𝑅1
37	27, 36	eqtri 2819	. . . . 5 ⊢ (𝑅1 “ On) = ran 𝑅1
38	26, 37	syl6eleqr 2887	. . . 4 ⊢ (𝐴 ∈ (𝑅1‘suc 𝑥) → (𝑅1‘suc 𝑥) ∈ (𝑅1 “ On))
39		elunii 4631	. . . 4 ⊢ ((𝐴 ∈ (𝑅1‘suc 𝑥) ∧ (𝑅1‘suc 𝑥) ∈ (𝑅1 “ On)) → 𝐴 ∈ ∪ (𝑅1 “ On))
40	38, 39	mpdan 679	. . 3 ⊢ (𝐴 ∈ (𝑅1‘suc 𝑥) → 𝐴 ∈ ∪ (𝑅1 “ On))
41	40	rexlimivw 3208	. 2 ⊢ (∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥) → 𝐴 ∈ ∪ (𝑅1 “ On))
42	23, 41	impbii 201	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   = wceq 1653  ∃wex 1875   ∈ wcel 2157  ∃wrex 3088   ⊆ wss 3767  𝒫 cpw 4347  ∪ cuni 4626  Tr wtr 4943  dom cdm 5310  ran crn 5311   ↾ cres 5312   “ cima 5313  Rel wrel 5315  Ord word 5938  Oncon0 5939  Lim wlim 5940  suc csuc 5941  Fun wfun 6093  ‘cfv 6099  𝑅₁cr1 8873

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-tr 4944  df-id 5218  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-we 5271  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-pred 5896  df-ord 5942  df-on 5943  df-lim 5944  df-suc 5945  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-om 7298  df-wrecs 7643  df-recs 7705  df-rdg 7743  df-r1 8875

This theorem is referenced by:  rankf  8905  r1elwf  8907  rankvalb  8908  rankidb  8911  rankwflem  8926  tcrank  8995  dfac12r  9254