Theorem onprcf1acwevdlem2 36087

Description: Lemma for onprcf1acwevd 36088 and prconf1acwevd 36089. (Contributed by BTernaryTau, 16-Sep-2026.)

Hypotheses

Ref	Expression
onprcf1acwevdlem2.1	⊢ 𝑅 = {⟨𝑦, 𝑧⟩ ∣ ((rank‘𝑦) ∈ (rank‘𝑧) ∨ ((rank‘𝑦) = (rank‘𝑧) ∧ 𝑦𝑆𝑧))}
onprcf1acwevdlem2.2	⊢ 𝑆 = (𝐹‘∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})
onprcf1acwevdlem2.3	⊢ ((𝜑 ∧ 𝑢 ∈ On) → ∃𝑤 ∈ On (𝐹‘𝑤) We (𝑅1‘𝑢))

Assertion

Ref	Expression
onprcf1acwevdlem2	⊢ (𝜑 → 𝑅 We V)

Distinct variable groups:   𝜑,𝑢,𝑦   𝑢,𝐹,𝑤   𝑧,𝑆   𝑦,𝐹,𝑧,𝑤

Allowed substitution hints:   𝜑(𝑧, 𝑤)   𝑅(𝑦, 𝑧, 𝑤, 𝑢)   𝑆(𝑦, 𝑤, 𝑢)

Proof of Theorem onprcf1acwevdlem2

Dummy variables 𝑞 𝑡 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rankon 9840	. . . . 5 ⊢ (rank‘𝑦) ∈ On
2	1	onsuci 7884	. . . 4 ⊢ suc (rank‘𝑦) ∈ On
3		onprcf1acwevdlem2.3	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ On) → ∃𝑤 ∈ On (𝐹‘𝑤) We (𝑅1‘𝑢))
4	3	ralrimiva 3158	. . . 4 ⊢ (𝜑 → ∀𝑢 ∈ On ∃𝑤 ∈ On (𝐹‘𝑤) We (𝑅1‘𝑢))
5		eqidd 2764	. . . . . . 7 ⊢ (𝑢 = suc (rank‘𝑦) → (𝐹‘𝑤) = (𝐹‘𝑤))
6		fveq2 6916	. . . . . . 7 ⊢ (𝑢 = suc (rank‘𝑦) → (𝑅1‘𝑢) = (𝑅1‘suc (rank‘𝑦)))
7	5, 6	weeq12d 5647	. . . . . 6 ⊢ (𝑢 = suc (rank‘𝑦) → ((𝐹‘𝑤) We (𝑅1‘𝑢) ↔ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))))
8	7	rexbidv 3190	. . . . 5 ⊢ (𝑢 = suc (rank‘𝑦) → (∃𝑤 ∈ On (𝐹‘𝑤) We (𝑅1‘𝑢) ↔ ∃𝑤 ∈ On (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))))
9	8	rspcv 3576	. . . 4 ⊢ (suc (rank‘𝑦) ∈ On → (∀𝑢 ∈ On ∃𝑤 ∈ On (𝐹‘𝑤) We (𝑅1‘𝑢) → ∃𝑤 ∈ On (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))))
10	2, 4, 9	mpsyl 69	. . 3 ⊢ (𝜑 → ∃𝑤 ∈ On (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦)))
11	10	alrimiv 1962	. 2 ⊢ (𝜑 → ∀𝑦∃𝑤 ∈ On (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦)))
12		vex 3458	. . . . . . . . 9 ⊢ 𝑣 ∈ V
13	12	rankr1 9883	. . . . . . . 8 ⊢ ((rank‘𝑦) = (rank‘𝑣) ↔ (¬ 𝑣 ∈ (𝑅1‘(rank‘𝑦)) ∧ 𝑣 ∈ (𝑅1‘suc (rank‘𝑦))))
14	13	simprbi 503	. . . . . . 7 ⊢ ((rank‘𝑦) = (rank‘𝑣) → 𝑣 ∈ (𝑅1‘suc (rank‘𝑦)))
15	14	eqcoms 2771	. . . . . 6 ⊢ ((rank‘𝑣) = (rank‘𝑦) → 𝑣 ∈ (𝑅1‘suc (rank‘𝑦)))
16	15	rgenw 3084	. . . . 5 ⊢ ∀𝑣 ∈ V ((rank‘𝑣) = (rank‘𝑦) → 𝑣 ∈ (𝑅1‘suc (rank‘𝑦)))
17		rabss 3998	. . . . 5 ⊢ ({𝑣 ∈ V ∣ (rank‘𝑣) = (rank‘𝑦)} ⊆ (𝑅1‘suc (rank‘𝑦)) ↔ ∀𝑣 ∈ V ((rank‘𝑣) = (rank‘𝑦) → 𝑣 ∈ (𝑅1‘suc (rank‘𝑦))))
18	16, 17	mpbir 234	. . . 4 ⊢ {𝑣 ∈ V ∣ (rank‘𝑣) = (rank‘𝑦)} ⊆ (𝑅1‘suc (rank‘𝑦))
19		onprcf1acwevdlem2.2	. . . . . . 7 ⊢ 𝑆 = (𝐹‘∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})
20		nfcv 2926	. . . . . . . 8 ⊢ Ⅎ𝑤𝐹
21		nfrab1 3435	. . . . . . . . 9 ⊢ Ⅎ𝑤{𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))}
22	21	nfint 4922	. . . . . . . 8 ⊢ Ⅎ𝑤∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))}
23	20, 22	nffv 6926	. . . . . . 7 ⊢ Ⅎ𝑤(𝐹‘∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})
24	19, 23	nfcxfr 2924	. . . . . 6 ⊢ Ⅎ𝑤𝑆
25		nfcv 2926	. . . . . 6 ⊢ Ⅎ𝑤(𝑅1‘suc (rank‘𝑦))
26	24, 25	nfwe 5633	. . . . 5 ⊢ Ⅎ𝑤 𝑆 We (𝑅1‘suc (rank‘𝑦))
27		fveq2 6916	. . . . . . 7 ⊢ (𝑤 = ∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))} → (𝐹‘𝑤) = (𝐹‘∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))}))
28	27, 19	eqtr4di 2816	. . . . . 6 ⊢ (𝑤 = ∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))} → (𝐹‘𝑤) = 𝑆)
29		eqidd 2764	. . . . . 6 ⊢ (𝑤 = ∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))} → (𝑅1‘suc (rank‘𝑦)) = (𝑅1‘suc (rank‘𝑦)))
30	28, 29	weeq12d 5647	. . . . 5 ⊢ (𝑤 = ∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))} → ((𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦)) ↔ 𝑆 We (𝑅1‘suc (rank‘𝑦))))
31	26, 30	onminsb 7842	. . . 4 ⊢ (∃𝑤 ∈ On (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦)) → 𝑆 We (𝑅1‘suc (rank‘𝑦)))
32		wess 5644	. . . 4 ⊢ ({𝑣 ∈ V ∣ (rank‘𝑣) = (rank‘𝑦)} ⊆ (𝑅1‘suc (rank‘𝑦)) → (𝑆 We (𝑅1‘suc (rank‘𝑦)) → 𝑆 We {𝑣 ∈ V ∣ (rank‘𝑣) = (rank‘𝑦)}))
33	18, 31, 32	mpsyl 69	. . 3 ⊢ (∃𝑤 ∈ On (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦)) → 𝑆 We {𝑣 ∈ V ∣ (rank‘𝑣) = (rank‘𝑦)})
34	33	alimi 1844	. 2 ⊢ (∀𝑦∃𝑤 ∈ On (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦)) → ∀𝑦 𝑆 We {𝑣 ∈ V ∣ (rank‘𝑣) = (rank‘𝑦)})
35		ralv 3480	. . 3 ⊢ (∀𝑦 ∈ V 𝑆 We {𝑣 ∈ V ∣ (rank‘𝑣) = (rank‘𝑦)} ↔ ∀𝑦 𝑆 We {𝑣 ∈ V ∣ (rank‘𝑣) = (rank‘𝑦)})
36		onprcf1acwevdlem2.1	. . . 4 ⊢ 𝑅 = {⟨𝑦, 𝑧⟩ ∣ ((rank‘𝑦) ∈ (rank‘𝑧) ∨ ((rank‘𝑦) = (rank‘𝑧) ∧ 𝑦𝑆𝑧))}
37		eqidd 2764	. . . . . . . . 9 ⊢ (𝑞 = (rank‘𝑦) → (𝐹‘𝑤) = (𝐹‘𝑤))
38		suceq 6457	. . . . . . . . . 10 ⊢ (𝑞 = (rank‘𝑦) → suc 𝑞 = suc (rank‘𝑦))
39	38	fveq2d 6920	. . . . . . . . 9 ⊢ (𝑞 = (rank‘𝑦) → (𝑅1‘suc 𝑞) = (𝑅1‘suc (rank‘𝑦)))
40	37, 39	weeq12d 5647	. . . . . . . 8 ⊢ (𝑞 = (rank‘𝑦) → ((𝐹‘𝑤) We (𝑅1‘suc 𝑞) ↔ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))))
41	40	rabbidv 3423	. . . . . . 7 ⊢ (𝑞 = (rank‘𝑦) → {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc 𝑞)} = {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})
42	41	inteqd 4917	. . . . . 6 ⊢ (𝑞 = (rank‘𝑦) → ∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc 𝑞)} = ∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})
43	42	fveq2d 6920	. . . . 5 ⊢ (𝑞 = (rank‘𝑦) → (𝐹‘∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc 𝑞)}) = (𝐹‘∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))}))
44	43, 19	eqtr4di 2816	. . . 4 ⊢ (𝑞 = (rank‘𝑦) → (𝐹‘∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc 𝑞)}) = 𝑆)
45		eqidd 2764	. . . . . . . . 9 ⊢ (𝑡 = 𝑦 → (𝐹‘𝑤) = (𝐹‘𝑤))
46		fveq2 6916	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑦 → (rank‘𝑡) = (rank‘𝑦))
47	46	suceqd 6456	. . . . . . . . . 10 ⊢ (𝑡 = 𝑦 → suc (rank‘𝑡) = suc (rank‘𝑦))
48	47	fveq2d 6920	. . . . . . . . 9 ⊢ (𝑡 = 𝑦 → (𝑅1‘suc (rank‘𝑡)) = (𝑅1‘suc (rank‘𝑦)))
49	45, 48	weeq12d 5647	. . . . . . . 8 ⊢ (𝑡 = 𝑦 → ((𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑡)) ↔ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))))
50	49	rabbidv 3423	. . . . . . 7 ⊢ (𝑡 = 𝑦 → {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑡))} = {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})
51	50	inteqd 4917	. . . . . 6 ⊢ (𝑡 = 𝑦 → ∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑡))} = ∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})
52	51	fveq2d 6920	. . . . 5 ⊢ (𝑡 = 𝑦 → (𝐹‘∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑡))}) = (𝐹‘∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))}))
53	52, 19	eqtr4di 2816	. . . 4 ⊢ (𝑡 = 𝑦 → (𝐹‘∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑡))}) = 𝑆)
54	36, 44, 53	werankwe 35862	. . 3 ⊢ (∀𝑦 ∈ V 𝑆 We {𝑣 ∈ V ∣ (rank‘𝑣) = (rank‘𝑦)} → 𝑅 We V)
55	35, 54	sylbir 238	. 2 ⊢ (∀𝑦 𝑆 We {𝑣 ∈ V ∣ (rank‘𝑣) = (rank‘𝑦)} → 𝑅 We V)
56	11, 34, 55	3syl 19	1 ⊢ (𝜑 → 𝑅 We V)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861  ∀wal 1568   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454   ⊆ wss 3900  ∩ cint 4912   class class class wbr 5108  {copab 5173   We wwe 5610  Oncon0 6388  suc csuc 6390  ‘cfv 6566  𝑅₁cr1 9803  rankcrnk 9804

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-reg 9623  ax-inf2 9679

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-r1 9805  df-rank 9806

This theorem is used by:  onprcf1acwevd  36088  prconf1acwevd  36089