Theorem onprcf1acwevdlem2 35896

Description: Lemma for onprcf1acwevd 35897. (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 9803	. . . . 5 ⊢ (rank‘𝑦) ∈ On
2	1	onsuci 7850	. . . 4 ⊢ suc (rank‘𝑦) ∈ On
3		onprcf1acwevdlem2.3	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ On) → ∃𝑤 ∈ On (𝐹‘𝑤) We (𝑅1‘𝑢))
4	3	ralrimiva 3155	. . . 4 ⊢ (𝜑 → ∀𝑢 ∈ On ∃𝑤 ∈ On (𝐹‘𝑤) We (𝑅1‘𝑢))
5		eqidd 2762	. . . . . . 7 ⊢ (𝑢 = suc (rank‘𝑦) → (𝐹‘𝑤) = (𝐹‘𝑤))
6		fveq2 6885	. . . . . . 7 ⊢ (𝑢 = suc (rank‘𝑦) → (𝑅1‘𝑢) = (𝑅1‘suc (rank‘𝑦)))
7	5, 6	weeq12d 5640	. . . . . 6 ⊢ (𝑢 = suc (rank‘𝑦) → ((𝐹‘𝑤) We (𝑅1‘𝑢) ↔ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))))
8	7	rexbidv 3187	. . . . 5 ⊢ (𝑢 = suc (rank‘𝑦) → (∃𝑤 ∈ On (𝐹‘𝑤) We (𝑅1‘𝑢) ↔ ∃𝑤 ∈ On (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))))
9	8	rspcv 3573	. . . 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 1960	. 2 ⊢ (𝜑 → ∀𝑦∃𝑤 ∈ On (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦)))
12		vex 3455	. . . . . . . . 9 ⊢ 𝑣 ∈ V
13	12	rankr1 9846	. . . . . . . 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 2769	. . . . . 6 ⊢ ((rank‘𝑣) = (rank‘𝑦) → 𝑣 ∈ (𝑅1‘suc (rank‘𝑦)))
16	15	rgenw 3081	. . . . 5 ⊢ ∀𝑣 ∈ V ((rank‘𝑣) = (rank‘𝑦) → 𝑣 ∈ (𝑅1‘suc (rank‘𝑦)))
17		rabss 4018	. . . . 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 2923	. . . . . . . 8 ⊢ Ⅎ𝑤𝐹
21		nfrab1 3432	. . . . . . . . 9 ⊢ Ⅎ𝑤{𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))}
22	21	nfint 4917	. . . . . . . 8 ⊢ Ⅎ𝑤∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))}
23	20, 22	nffv 6895	. . . . . . 7 ⊢ Ⅎ𝑤(𝐹‘∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})
24	19, 23	nfcxfr 2921	. . . . . 6 ⊢ Ⅎ𝑤𝑆
25		nfcv 2923	. . . . . 6 ⊢ Ⅎ𝑤(𝑅1‘suc (rank‘𝑦))
26	24, 25	nfwe 5626	. . . . 5 ⊢ Ⅎ𝑤 𝑆 We (𝑅1‘suc (rank‘𝑦))
27		fveq2 6885	. . . . . . 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 2814	. . . . . 6 ⊢ (𝑤 = ∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))} → (𝐹‘𝑤) = 𝑆)
29		eqidd 2762	. . . . . 6 ⊢ (𝑤 = ∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))} → (𝑅1‘suc (rank‘𝑦)) = (𝑅1‘suc (rank‘𝑦)))
30	28, 29	weeq12d 5640	. . . . 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 7808	. . . 4 ⊢ (∃𝑤 ∈ On (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦)) → 𝑆 We (𝑅1‘suc (rank‘𝑦)))
32		wess 5637	. . . 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 3477	. . 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 2762	. . . . . . . . 9 ⊢ (𝑞 = (rank‘𝑦) → (𝐹‘𝑤) = (𝐹‘𝑤))
38		suceq 6431	. . . . . . . . . 10 ⊢ (𝑞 = (rank‘𝑦) → suc 𝑞 = suc (rank‘𝑦))
39	38	fveq2d 6889	. . . . . . . . 9 ⊢ (𝑞 = (rank‘𝑦) → (𝑅1‘suc 𝑞) = (𝑅1‘suc (rank‘𝑦)))
40	37, 39	weeq12d 5640	. . . . . . . 8 ⊢ (𝑞 = (rank‘𝑦) → ((𝐹‘𝑤) We (𝑅1‘suc 𝑞) ↔ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))))
41	40	rabbidv 3420	. . . . . . 7 ⊢ (𝑞 = (rank‘𝑦) → {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc 𝑞)} = {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})
42	41	inteqd 4912	. . . . . 6 ⊢ (𝑞 = (rank‘𝑦) → ∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc 𝑞)} = ∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})
43	42	fveq2d 6889	. . . . 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 2814	. . . 4 ⊢ (𝑞 = (rank‘𝑦) → (𝐹‘∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc 𝑞)}) = 𝑆)
45		eqidd 2762	. . . . . . . . 9 ⊢ (𝑡 = 𝑦 → (𝐹‘𝑤) = (𝐹‘𝑤))
46		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑦 → (rank‘𝑡) = (rank‘𝑦))
47	46	suceqd 6430	. . . . . . . . . 10 ⊢ (𝑡 = 𝑦 → suc (rank‘𝑡) = suc (rank‘𝑦))
48	47	fveq2d 6889	. . . . . . . . 9 ⊢ (𝑡 = 𝑦 → (𝑅1‘suc (rank‘𝑡)) = (𝑅1‘suc (rank‘𝑦)))
49	45, 48	weeq12d 5640	. . . . . . . 8 ⊢ (𝑡 = 𝑦 → ((𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑡)) ↔ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))))
50	49	rabbidv 3420	. . . . . . 7 ⊢ (𝑡 = 𝑦 → {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑡))} = {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})
51	50	inteqd 4912	. . . . . 6 ⊢ (𝑡 = 𝑦 → ∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑡))} = ∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})
52	51	fveq2d 6889	. . . . 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 2814	. . . 4 ⊢ (𝑡 = 𝑦 → (𝐹‘∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑡))}) = 𝑆)
54	36, 44, 53	werankwe 35739	. . 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 2145  ∀wral 3077  ∃wrex 3087  {crab 3413  Vcvv 3451   ⊆ wss 3899  ∩ cint 4907   class class class wbr 5103  {copab 5167   We wwe 5603  Oncon0 6362  suc csuc 6364  ‘cfv 6538  𝑅₁cr1 9766  rankcrnk 9767

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-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-reg 9586  ax-inf2 9642

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-tp 4589  df-op 4591  df-uni 4868  df-int 4908  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 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-r1 9768  df-rank 9769

This theorem is used by:  onprcf1acwevd  35897