Theorem rankwflemb 9707

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 4865	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ (𝑅1 “ On)))
2		eleq2 2824	. . . . . . . 8 ⊢ ((𝑅1‘𝑥) = 𝑦 → (𝐴 ∈ (𝑅1‘𝑥) ↔ 𝐴 ∈ 𝑦))
3	2	biimprcd 250	. . . . . . 7 ⊢ (𝐴 ∈ 𝑦 → ((𝑅1‘𝑥) = 𝑦 → 𝐴 ∈ (𝑅1‘𝑥)))
4		r1tr 9690	. . . . . . . . . . 11 ⊢ Tr (𝑅1‘𝑥)
5		trss 5214	. . . . . . . . . . 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 4556	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝑅1‘𝑥) → (𝐴 ∈ 𝒫 (𝑅1‘𝑥) ↔ 𝐴 ⊆ (𝑅1‘𝑥)))
8	6, 7	mpbird 257	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑅1‘𝑥) → 𝐴 ∈ 𝒫 (𝑅1‘𝑥))
9		elfvdm 6867	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝑅1‘𝑥) → 𝑥 ∈ dom 𝑅1)
10		r1sucg 9683	. . . . . . . . . 10 ⊢ (𝑥 ∈ dom 𝑅1 → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
11	9, 10	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑅1‘𝑥) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
12	8, 11	eleqtrrd 2838	. . . . . . . 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 77	. . . . . 6 ⊢ (𝐴 ∈ 𝑦 → (𝑥 ∈ On → ((𝑅1‘𝑥) = 𝑦 → 𝐴 ∈ (𝑅1‘suc 𝑥))))
15	14	reximdvai 3146	. . . . 5 ⊢ (𝐴 ∈ 𝑦 → (∃𝑥 ∈ On (𝑅1‘𝑥) = 𝑦 → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥)))
16		r1funlim 9680	. . . . . . 7 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
17	16	simpli 483	. . . . . 6 ⊢ Fun 𝑅1
18		fvelima 6898	. . . . . 6 ⊢ ((Fun 𝑅1 ∧ 𝑦 ∈ (𝑅1 “ On)) → ∃𝑥 ∈ On (𝑅1‘𝑥) = 𝑦)
19	17, 18	mpan 691	. . . . 5 ⊢ (𝑦 ∈ (𝑅1 “ On) → ∃𝑥 ∈ On (𝑅1‘𝑥) = 𝑦)
20	15, 19	impel 505	. . . 4 ⊢ ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ (𝑅1 “ On)) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
21	20	exlimiv 1932	. . 3 ⊢ (∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ (𝑅1 “ On)) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
22	1, 21	sylbi 217	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
23		elfvdm 6867	. . . . . 6 ⊢ (𝐴 ∈ (𝑅1‘suc 𝑥) → suc 𝑥 ∈ dom 𝑅1)
24		fvelrn 7021	. . . . . 6 ⊢ ((Fun 𝑅1 ∧ suc 𝑥 ∈ dom 𝑅1) → (𝑅1‘suc 𝑥) ∈ ran 𝑅1)
25	17, 23, 24	sylancr 588	. . . . 5 ⊢ (𝐴 ∈ (𝑅1‘suc 𝑥) → (𝑅1‘suc 𝑥) ∈ ran 𝑅1)
26		df-ima 5636	. . . . . 6 ⊢ (𝑅1 “ On) = ran (𝑅1 ↾ On)
27		funrel 6508	. . . . . . . . 9 ⊢ (Fun 𝑅1 → Rel 𝑅1)
28	17, 27	ax-mp 5	. . . . . . . 8 ⊢ Rel 𝑅1
29	16	simpri 485	. . . . . . . . 9 ⊢ Lim dom 𝑅1
30		limord 6377	. . . . . . . . 9 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
31		ordsson 7728	. . . . . . . . 9 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
32	29, 30, 31	mp2b 10	. . . . . . . 8 ⊢ dom 𝑅1 ⊆ On
33		relssres 5980	. . . . . . . 8 ⊢ ((Rel 𝑅1 ∧ dom 𝑅1 ⊆ On) → (𝑅1 ↾ On) = 𝑅1)
34	28, 32, 33	mp2an 693	. . . . . . 7 ⊢ (𝑅1 ↾ On) = 𝑅1
35	34	rneqi 5885	. . . . . 6 ⊢ ran (𝑅1 ↾ On) = ran 𝑅1
36	26, 35	eqtri 2758	. . . . 5 ⊢ (𝑅1 “ On) = ran 𝑅1
37	25, 36	eleqtrrdi 2846	. . . 4 ⊢ (𝐴 ∈ (𝑅1‘suc 𝑥) → (𝑅1‘suc 𝑥) ∈ (𝑅1 “ On))
38		elunii 4867	. . . 4 ⊢ ((𝐴 ∈ (𝑅1‘suc 𝑥) ∧ (𝑅1‘suc 𝑥) ∈ (𝑅1 “ On)) → 𝐴 ∈ ∪ (𝑅1 “ On))
39	37, 38	mpdan 688	. . 3 ⊢ (𝐴 ∈ (𝑅1‘suc 𝑥) → 𝐴 ∈ ∪ (𝑅1 “ On))
40	39	rexlimivw 3132	. 2 ⊢ (∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥) → 𝐴 ∈ ∪ (𝑅1 “ On))
41	22, 40	impbii 209	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃wrex 3059   ⊆ wss 3900  𝒫 cpw 4553  ∪ cuni 4862  Tr wtr 5204  dom cdm 5623  ran crn 5624   ↾ cres 5625   “ cima 5626  Rel wrel 5628  Ord word 6315  Oncon0 6316  Lim wlim 6317  suc csuc 6318  Fun wfun 6485  ‘cfv 6491  𝑅₁cr1 9676

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 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-r1 9678

This theorem is referenced by:  rankf  9708  r1elwf  9710  rankvalb  9711  rankidb  9714  rankwflem  9729  tcrank  9798  dfac12r  10059