Theorem fineqvrep 35109

Description: If the Axiom of Infinity is negated, 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 6601	. . . 4 ⊢ (Fun {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ↔ ∀𝑤∃*𝑧∀𝑦𝜑)
2		nfa1 2151	. . . . . 6 ⊢ Ⅎ𝑦∀𝑦𝜑
3	2	mof 2563	. . . . 5 ⊢ (∃*𝑧∀𝑦𝜑 ↔ ∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦))
4	3	albii 1819	. . . 4 ⊢ (∀𝑤∃*𝑧∀𝑦𝜑 ↔ ∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦))
5	1, 4	bitr2i 276	. . 3 ⊢ (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) ↔ Fun {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
6		vex 3484	. . . . . . 7 ⊢ 𝑥 ∈ V
7		eleq2w2 2733	. . . . . . 7 ⊢ (Fin = V → (𝑥 ∈ Fin ↔ 𝑥 ∈ V))
8	6, 7	mpbiri 258	. . . . . 6 ⊢ (Fin = V → 𝑥 ∈ Fin)
9		imafi 9353	. . . . . 6 ⊢ ((Fun {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ∧ 𝑥 ∈ Fin) → ({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) ∈ Fin)
10	8, 9	sylan2 593	. . . . 5 ⊢ ((Fun {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ∧ Fin = V) → ({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) ∈ Fin)
11	10	elexd 3504	. . . 4 ⊢ ((Fun {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ∧ Fin = V) → ({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) ∈ V)
12		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑦 𝑤 ∈ 𝑥
13	2	nfopab 5212	. . . . . . . . . . 11 ⊢ Ⅎ𝑦{⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
14	13	nfel2 2924	. . . . . . . . . 10 ⊢ Ⅎ𝑦⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
15	12, 14	nfan 1899	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
16	15	nfex 2324	. . . . . . . 8 ⊢ Ⅎ𝑦∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
17	16	nfab 2911	. . . . . . 7 ⊢ Ⅎ𝑦{𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})}
18	17	issetf 3497	. . . . . 6 ⊢ ({𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})} ∈ V ↔ ∃𝑦 𝑦 = {𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})})
19		eqabb 2881	. . . . . . 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 1848	. . . . . 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 5529	. . . . . . . . . . 11 ⊢ (⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ↔ ∀𝑦𝜑)
22	21	anbi2i 623	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))
23	22	exbii 1848	. . . . . . . . 9 ⊢ (∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))
24	23	bibi2i 337	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})) ↔ (𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
25	24	albii 1819	. . . . . . 7 ⊢ (∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})) ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
26	25	exbii 1848	. . . . . 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 298	. . . . 5 ⊢ (∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)) ↔ {𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})} ∈ V)
28		dfima3 6081	. . . . . . 7 ⊢ ({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) = {𝑣 ∣ ∃𝑢(𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})}
29		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑧 𝑢 ∈ 𝑥
30		nfopab2 5214	. . . . . . . . . . 11 ⊢ Ⅎ𝑧{⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
31	30	nfel2 2924	. . . . . . . . . 10 ⊢ Ⅎ𝑧⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
32	29, 31	nfan 1899	. . . . . . . . 9 ⊢ Ⅎ𝑧(𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
33	32	nfex 2324	. . . . . . . 8 ⊢ Ⅎ𝑧∃𝑢(𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
34		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑣∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
35		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑤 𝑢 ∈ 𝑥
36		nfopab1 5213	. . . . . . . . . . . 12 ⊢ Ⅎ𝑤{⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
37	36	nfel2 2924	. . . . . . . . . . 11 ⊢ Ⅎ𝑤⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
38	35, 37	nfan 1899	. . . . . . . . . 10 ⊢ Ⅎ𝑤(𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
39		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑢(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
40		elequ1 2115	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑤 → (𝑢 ∈ 𝑥 ↔ 𝑤 ∈ 𝑥))
41		opeq1 4873	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑤 → ⟨𝑢, 𝑣⟩ = ⟨𝑤, 𝑣⟩)
42	41	eleq1d 2826	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑤 → (⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ↔ ⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}))
43	40, 42	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑢 = 𝑤 → ((𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ (𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})))
44	38, 39, 43	cbvexv1 2344	. . . . . . . . 9 ⊢ (∃𝑢(𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}))
45		opeq2 4874	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑧 → ⟨𝑤, 𝑣⟩ = ⟨𝑤, 𝑧⟩)
46	45	eleq1d 2826	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑧 → (⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ↔ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}))
47	46	anbi2d 630	. . . . . . . . . 10 ⊢ (𝑣 = 𝑧 → ((𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ (𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})))
48	47	exbidv 1921	. . . . . . . . 9 ⊢ (𝑣 = 𝑧 → (∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})))
49	44, 48	bitrid 283	. . . . . . . 8 ⊢ (𝑣 = 𝑧 → (∃𝑢(𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})))
50	33, 34, 49	cbvabw 2813	. . . . . . 7 ⊢ {𝑣 ∣ ∃𝑢(𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})} = {𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})}
51	28, 50	eqtri 2765	. . . . . 6 ⊢ ({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) = {𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})}
52	51	eleq1i 2832	. . . . 5 ⊢ (({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) ∈ V ↔ {𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})} ∈ V)
53	27, 52	bitr4i 278	. . . 4 ⊢ (∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)) ↔ ({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) ∈ V)
54	11, 53	sylibr 234	. . 3 ⊢ ((Fun {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ∧ Fin = V) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
55	5, 54	sylanb 581	. 2 ⊢ ((∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) ∧ Fin = V) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
56	55	expcom 413	1 ⊢ (Fin = V → (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∃*wmo 2538  {cab 2714  Vcvv 3480  ⟨cop 4632  {copab 5205   “ cima 5688  Fun wfun 6555  Fincfn 8985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-en 8986  df-dom 8987  df-fin 8989

This theorem is referenced by: (None)