Theorem fineqvrep 35782

Description: If all sets are finite, then the Axiom of Replacement becomes redundant. (Contributed by BTernaryTau, 12-Sep-2024.)

Assertion

Ref	Expression
fineqvrep	⊢ (Fin = V → (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))

Distinct variable group:   𝑥,𝑤,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧, 𝑤)

Proof of Theorem fineqvrep

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funopab 6575	. . . 4 ⊢ (Fun {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ↔ ∀𝑤∃*𝑧∀𝑦𝜑)
2		nfa1 2188	. . . . . 6 ⊢ Ⅎ𝑦∀𝑦𝜑
3	2	mof 2589	. . . . 5 ⊢ (∃*𝑧∀𝑦𝜑 ↔ ∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦))
4	3	albii 1852	. . . 4 ⊢ (∀𝑤∃*𝑧∀𝑦𝜑 ↔ ∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦))
5	1, 4	bitr2i 279	. . 3 ⊢ (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) ↔ Fun {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
6		vex 3455	. . . . . . 7 ⊢ 𝑥 ∈ V
7		eleq2w2 2757	. . . . . . 7 ⊢ (Fin = V → (𝑥 ∈ Fin ↔ 𝑥 ∈ V))
8	6, 7	mpbiri 261	. . . . . 6 ⊢ (Fin = V → 𝑥 ∈ Fin)
9		imafi 9307	. . . . . 6 ⊢ ((Fun {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ∧ 𝑥 ∈ Fin) → ({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) ∈ Fin)
10	8, 9	sylan2 605	. . . . 5 ⊢ ((Fun {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ∧ Fin = V) → ({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) ∈ Fin)
11	10	elexd 3474	. . . 4 ⊢ ((Fun {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ∧ Fin = V) → ({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) ∈ V)
12		nfv 1947	. . . . . . . . . 10 ⊢ Ⅎ𝑦 𝑤 ∈ 𝑥
13	2	nfopab 5174	. . . . . . . . . . 11 ⊢ Ⅎ𝑦{⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
14	13	nfel2 2941	. . . . . . . . . 10 ⊢ Ⅎ𝑦⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
15	12, 14	nfan 1932	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
16	15	nfex 2355	. . . . . . . 8 ⊢ Ⅎ𝑦∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
17	16	nfab 2929	. . . . . . 7 ⊢ Ⅎ𝑦{𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})}
18	17	issetf 3468	. . . . . 6 ⊢ ({𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})} ∈ V ↔ ∃𝑦 𝑦 = {𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})})
19		eqabb 2900	. . . . . . 7 ⊢ (𝑦 = {𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})} ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})))
20	19	exbii 1881	. . . . . 6 ⊢ (∃𝑦 𝑦 = {𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})} ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})))
21		opabidw 5498	. . . . . . . . . . 11 ⊢ (⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ↔ ∀𝑦𝜑)
22	21	anbi2i 635	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))
23	22	exbii 1881	. . . . . . . . 9 ⊢ (∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))
24	23	bibi2i 340	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})) ↔ (𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
25	24	albii 1852	. . . . . . 7 ⊢ (∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})) ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
26	25	exbii 1881	. . . . . 6 ⊢ (∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})) ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
27	18, 20, 26	3bitrri 301	. . . . 5 ⊢ (∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)) ↔ {𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})} ∈ V)
28		dfima3 6059	. . . . . . 7 ⊢ ({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) = {𝑣 ∣ ∃𝑢(𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})}
29		nfv 1947	. . . . . . . . . 10 ⊢ Ⅎ𝑧 𝑢 ∈ 𝑥
30		nfopab2 5176	. . . . . . . . . . 11 ⊢ Ⅎ𝑧{⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
31	30	nfel2 2941	. . . . . . . . . 10 ⊢ Ⅎ𝑧⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
32	29, 31	nfan 1932	. . . . . . . . 9 ⊢ Ⅎ𝑧(𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
33	32	nfex 2355	. . . . . . . 8 ⊢ Ⅎ𝑧∃𝑢(𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
34		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑣∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
35		nfv 1947	. . . . . . . . . . 11 ⊢ Ⅎ𝑤 𝑢 ∈ 𝑥
36		nfopab1 5175	. . . . . . . . . . . 12 ⊢ Ⅎ𝑤{⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
37	36	nfel2 2941	. . . . . . . . . . 11 ⊢ Ⅎ𝑤⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
38	35, 37	nfan 1932	. . . . . . . . . 10 ⊢ Ⅎ𝑤(𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
39		nfv 1947	. . . . . . . . . 10 ⊢ Ⅎ𝑢(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
40		elequ1 2152	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑤 → (𝑢 ∈ 𝑥 ↔ 𝑤 ∈ 𝑥))
41		opeq1 4833	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑤 → ⟨𝑢, 𝑣⟩ = ⟨𝑤, 𝑣⟩)
42	41	eleq1d 2846	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑤 → (⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ↔ ⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}))
43	40, 42	anbi12d 644	. . . . . . . . . 10 ⊢ (𝑢 = 𝑤 → ((𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ (𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})))
44	38, 39, 43	cbvexv1 2372	. . . . . . . . 9 ⊢ (∃𝑢(𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}))
45		opeq2 4834	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑧 → ⟨𝑤, 𝑣⟩ = ⟨𝑤, 𝑧⟩)
46	45	eleq1d 2846	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑧 → (⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ↔ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}))
47	46	anbi2d 642	. . . . . . . . . 10 ⊢ (𝑣 = 𝑧 → ((𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ (𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})))
48	47	exbidv 1954	. . . . . . . . 9 ⊢ (𝑣 = 𝑧 → (∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})))
49	44, 48	bitrid 286	. . . . . . . 8 ⊢ (𝑣 = 𝑧 → (∃𝑢(𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})))
50	33, 34, 49	cbvabw 2832	. . . . . . 7 ⊢ {𝑣 ∣ ∃𝑢(𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})} = {𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})}
51	28, 50	eqtri 2784	. . . . . 6 ⊢ ({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) = {𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})}
52	51	eleq1i 2852	. . . . 5 ⊢ (({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) ∈ V ↔ {𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})} ∈ V)
53	27, 52	bitr4i 281	. . . 4 ⊢ (∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)) ↔ ({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) ∈ V)
54	11, 53	sylibr 237	. . 3 ⊢ ((Fun {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ∧ Fin = V) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
55	5, 54	sylanb 593	. 2 ⊢ ((∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) ∧ Fin = V) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
56	55	expcom 419	1 ⊢ (Fin = V → (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃*wmo 2563  {cab 2739  Vcvv 3451  ⟨cop 4590  {copab 5167   “ cima 5654  Fun wfun 6532  Fincfn 8973

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-pr 5391  ax-un 7751

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-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-br 5104  df-opab 5168  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-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-om 7878  df-1o 8476  df-en 8974  df-dom 8975  df-fin 8977

This theorem is used by: (None)