Theorem rankwflembOLD 9794

Description: Obsolete version of rankwflemb 9793 as of 29-Sep-2026. (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Proof of Theorem rankwflembOLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni 4870	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ (𝑅1 “ On)))
2		eleq2 2850	. . . . . . . 8 ⊢ ((𝑅1‘𝑥) = 𝑦 → (𝐴 ∈ (𝑅1‘𝑥) ↔ 𝐴 ∈ 𝑦))
3	2	biimprcd 253	. . . . . . 7 ⊢ (𝐴 ∈ 𝑦 → ((𝑅1‘𝑥) = 𝑦 → 𝐴 ∈ (𝑅1‘𝑥)))
4		r1tr 9776	. . . . . . . . . . 11 ⊢ Tr (𝑅1‘𝑥)
5		trss 5222	. . . . . . . . . . 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 4560	. . . . . . . . . 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 6917	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝑅1‘𝑥) → 𝑥 ∈ dom 𝑅1)
10		r1sucg 9769	. . . . . . . . . 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 2864	. . . . . . . 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 3174	. . . . 5 ⊢ (𝐴 ∈ 𝑦 → (∃𝑥 ∈ On (𝑅1‘𝑥) = 𝑦 → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥)))
16		r1fun 9764	. . . . . 6 ⊢ Fun 𝑅1
17		fvelima 6948	. . . . . 6 ⊢ ((Fun 𝑅1 ∧ 𝑦 ∈ (𝑅1 “ On)) → ∃𝑥 ∈ On (𝑅1‘𝑥) = 𝑦)
18	16, 17	mpan 703	. . . . 5 ⊢ (𝑦 ∈ (𝑅1 “ On) → ∃𝑥 ∈ On (𝑅1‘𝑥) = 𝑦)
19	15, 18	impel 515	. . . 4 ⊢ ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ (𝑅1 “ On)) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
20	19	exlimiv 1963	. . 3 ⊢ (∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ (𝑅1 “ On)) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
21	1, 20	sylbi 220	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
22		elfvdm 6917	. . . . . 6 ⊢ (𝐴 ∈ (𝑅1‘suc 𝑥) → suc 𝑥 ∈ dom 𝑅1)
23		fvelrn 7074	. . . . . 6 ⊢ ((Fun 𝑅1 ∧ suc 𝑥 ∈ dom 𝑅1) → (𝑅1‘suc 𝑥) ∈ ran 𝑅1)
24	16, 22, 23	sylancr 599	. . . . 5 ⊢ (𝐴 ∈ (𝑅1‘suc 𝑥) → (𝑅1‘suc 𝑥) ∈ ran 𝑅1)
25		df-ima 5664	. . . . . 6 ⊢ (𝑅1 “ On) = ran (𝑅1 ↾ On)
26		funrel 6554	. . . . . . . . 9 ⊢ (Fun 𝑅1 → Rel 𝑅1)
27	16, 26	ax-mp 5	. . . . . . . 8 ⊢ Rel 𝑅1
28		r1dmlim 9765	. . . . . . . . 9 ⊢ Lim dom 𝑅1
29		limord 6423	. . . . . . . . 9 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
30		ordsson 7795	. . . . . . . . 9 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
31	28, 29, 30	mp2b 10	. . . . . . . 8 ⊢ dom 𝑅1 ⊆ On
32		relssres 6011	. . . . . . . 8 ⊢ ((Rel 𝑅1 ∧ dom 𝑅1 ⊆ On) → (𝑅1 ↾ On) = 𝑅1)
33	27, 31, 32	mp2an 705	. . . . . . 7 ⊢ (𝑅1 ↾ On) = 𝑅1
34	33	rneqi 5919	. . . . . 6 ⊢ ran (𝑅1 ↾ On) = ran 𝑅1
35	25, 34	eqtri 2784	. . . . 5 ⊢ (𝑅1 “ On) = ran 𝑅1
36	24, 35	eleqtrrdi 2872	. . . 4 ⊢ (𝐴 ∈ (𝑅1‘suc 𝑥) → (𝑅1‘suc 𝑥) ∈ (𝑅1 “ On))
37		elunii 4872	. . . 4 ⊢ ((𝐴 ∈ (𝑅1‘suc 𝑥) ∧ (𝑅1‘suc 𝑥) ∈ (𝑅1 “ On)) → 𝐴 ∈ ∪ (𝑅1 “ On))
38	36, 37	mpdan 700	. . 3 ⊢ (𝐴 ∈ (𝑅1‘suc 𝑥) → 𝐴 ∈ ∪ (𝑅1 “ On))
39	38	rexlimivw 3160	. 2 ⊢ (∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥) → 𝐴 ∈ ∪ (𝑅1 “ On))
40	21, 39	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 3087   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867  Tr wtr 5212  dom cdm 5651  ran crn 5652   ↾ cres 5653   “ cima 5654  Rel wrel 5656  Ord word 6360  Oncon0 6361  Lim wlim 6362  suc csuc 6363  Fun wfun 6531  ‘cfv 6537  𝑅₁cr1 9759

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 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  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 7421  df-om 7876  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-r1 9761

This theorem is used by: (None)