Theorem fineqvrep 35527

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 6571	. . . 4 ⊢ (Fun {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ↔ ∀𝑤∃*𝑧∀𝑦𝜑)
2		nfa1 2186	. . . . . 6 ⊢ Ⅎ𝑦∀𝑦𝜑
3	2	mof 2591	. . . . 5 ⊢ (∃*𝑧∀𝑦𝜑 ↔ ∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦))
4	3	albii 1849	. . . 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 3459	. . . . . . 7 ⊢ 𝑥 ∈ V
7		eleq2w2 2759	. . . . . . 7 ⊢ (Fin = V → (𝑥 ∈ Fin ↔ 𝑥 ∈ V))
8	6, 7	mpbiri 261	. . . . . 6 ⊢ (Fin = V → 𝑥 ∈ Fin)
9		imafi 9271	. . . . . 6 ⊢ ((Fun {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ∧ 𝑥 ∈ Fin) → ({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) ∈ Fin)
10	8, 9	sylan2 604	. . . . 5 ⊢ ((Fun {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ∧ Fin = V) → ({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) ∈ Fin)
11	10	elexd 3478	. . . 4 ⊢ ((Fun {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ∧ Fin = V) → ({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) ∈ V)
12		nfv 1944	. . . . . . . . . 10 ⊢ Ⅎ𝑦 𝑤 ∈ 𝑥
13	2	nfopab 5180	. . . . . . . . . . 11 ⊢ Ⅎ𝑦{⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
14	13	nfel2 2943	. . . . . . . . . 10 ⊢ Ⅎ𝑦⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
15	12, 14	nfan 1929	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
16	15	nfex 2357	. . . . . . . 8 ⊢ Ⅎ𝑦∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
17	16	nfab 2931	. . . . . . 7 ⊢ Ⅎ𝑦{𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})}
18	17	issetf 3472	. . . . . 6 ⊢ ({𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})} ∈ V ↔ ∃𝑦 𝑦 = {𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})})
19		eqabb 2902	. . . . . . 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 1878	. . . . . 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 5508	. . . . . . . . . . 11 ⊢ (⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ↔ ∀𝑦𝜑)
22	21	anbi2i 634	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))
23	22	exbii 1878	. . . . . . . . 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 1849	. . . . . . 7 ⊢ (∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})) ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
26	25	exbii 1878	. . . . . 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 6065	. . . . . . 7 ⊢ ({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) = {𝑣 ∣ ∃𝑢(𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})}
29		nfv 1944	. . . . . . . . . 10 ⊢ Ⅎ𝑧 𝑢 ∈ 𝑥
30		nfopab2 5182	. . . . . . . . . . 11 ⊢ Ⅎ𝑧{⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
31	30	nfel2 2943	. . . . . . . . . 10 ⊢ Ⅎ𝑧⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
32	29, 31	nfan 1929	. . . . . . . . 9 ⊢ Ⅎ𝑧(𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
33	32	nfex 2357	. . . . . . . 8 ⊢ Ⅎ𝑧∃𝑢(𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
34		nfv 1944	. . . . . . . 8 ⊢ Ⅎ𝑣∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
35		nfv 1944	. . . . . . . . . . 11 ⊢ Ⅎ𝑤 𝑢 ∈ 𝑥
36		nfopab1 5181	. . . . . . . . . . . 12 ⊢ Ⅎ𝑤{⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
37	36	nfel2 2943	. . . . . . . . . . 11 ⊢ Ⅎ𝑤⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
38	35, 37	nfan 1929	. . . . . . . . . 10 ⊢ Ⅎ𝑤(𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
39		nfv 1944	. . . . . . . . . 10 ⊢ Ⅎ𝑢(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
40		elequ1 2150	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑤 → (𝑢 ∈ 𝑥 ↔ 𝑤 ∈ 𝑥))
41		opeq1 4838	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑤 → ⟨𝑢, 𝑣⟩ = ⟨𝑤, 𝑣⟩)
42	41	eleq1d 2848	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑤 → (⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ↔ ⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}))
43	40, 42	anbi12d 643	. . . . . . . . . 10 ⊢ (𝑢 = 𝑤 → ((𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ (𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})))
44	38, 39, 43	cbvexv1 2374	. . . . . . . . 9 ⊢ (∃𝑢(𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}))
45		opeq2 4839	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑧 → ⟨𝑤, 𝑣⟩ = ⟨𝑤, 𝑧⟩)
46	45	eleq1d 2848	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑧 → (⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ↔ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}))
47	46	anbi2d 641	. . . . . . . . . 10 ⊢ (𝑣 = 𝑧 → ((𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ (𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})))
48	47	exbidv 1951	. . . . . . . . 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 2834	. . . . . . 7 ⊢ {𝑣 ∣ ∃𝑢(𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})} = {𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})}
51	28, 50	eqtri 2786	. . . . . 6 ⊢ ({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) = {𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})}
52	51	eleq1i 2854	. . . . 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 592	. 2 ⊢ ((∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) ∧ Fin = V) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
56	55	expcom 418	1 ⊢ (Fin = V → (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1568   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∃*wmo 2565  {cab 2741  Vcvv 3455  ⟨cop 4595  {copab 5173   “ cima 5664  Fun wfun 6530  Fincfn 8939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-om 7859  df-1o 8449  df-en 8940  df-dom 8941  df-fin 8943

This theorem is referenced by: (None)