Theorem fineqvrep 32647

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 6384	. . . 4 ⊢ (Fun {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ↔ ∀𝑤∃*𝑧∀𝑦𝜑)
2		nfa1 2156	. . . . . 6 ⊢ Ⅎ𝑦∀𝑦𝜑
3	2	mof 2564	. . . . 5 ⊢ (∃*𝑧∀𝑦𝜑 ↔ ∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦))
4	3	albii 1826	. . . 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 3404	. . . . . . 7 ⊢ 𝑥 ∈ V
7		eleq2w2 2735	. . . . . . 7 ⊢ (Fin = V → (𝑥 ∈ Fin ↔ 𝑥 ∈ V))
8	6, 7	mpbiri 261	. . . . . 6 ⊢ (Fin = V → 𝑥 ∈ Fin)
9		imafi 8785	. . . . . 6 ⊢ ((Fun {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ∧ 𝑥 ∈ Fin) → ({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) ∈ Fin)
10	8, 9	sylan2 596	. . . . 5 ⊢ ((Fun {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ∧ Fin = V) → ({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) ∈ Fin)
11	10	elexd 3420	. . . 4 ⊢ ((Fun {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ∧ Fin = V) → ({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) ∈ V)
12		nfv 1921	. . . . . . . . . 10 ⊢ Ⅎ𝑦 𝑤 ∈ 𝑥
13	2	nfopab 5108	. . . . . . . . . . 11 ⊢ Ⅎ𝑦{⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
14	13	nfel2 2918	. . . . . . . . . 10 ⊢ Ⅎ𝑦⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
15	12, 14	nfan 1906	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
16	15	nfex 2327	. . . . . . . 8 ⊢ Ⅎ𝑦∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
17	16	nfab 2906	. . . . . . 7 ⊢ Ⅎ𝑦{𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})}
18	17	issetf 3414	. . . . . 6 ⊢ ({𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})} ∈ V ↔ ∃𝑦 𝑦 = {𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})})
19		abeq2 2865	. . . . . . 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 1854	. . . . . 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 5390	. . . . . . . . . . 11 ⊢ (⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ↔ ∀𝑦𝜑)
22	21	anbi2i 626	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))
23	22	exbii 1854	. . . . . . . . 9 ⊢ (∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))
24	23	bibi2i 341	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})) ↔ (𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
25	24	albii 1826	. . . . . . 7 ⊢ (∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})) ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
26	25	exbii 1854	. . . . . 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 5916	. . . . . . 7 ⊢ ({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) = {𝑣 ∣ ∃𝑢(𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})}
29		nfv 1921	. . . . . . . . . 10 ⊢ Ⅎ𝑧 𝑢 ∈ 𝑥
30		nfopab2 5110	. . . . . . . . . . 11 ⊢ Ⅎ𝑧{⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
31	30	nfel2 2918	. . . . . . . . . 10 ⊢ Ⅎ𝑧⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
32	29, 31	nfan 1906	. . . . . . . . 9 ⊢ Ⅎ𝑧(𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
33	32	nfex 2327	. . . . . . . 8 ⊢ Ⅎ𝑧∃𝑢(𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
34		nfv 1921	. . . . . . . 8 ⊢ Ⅎ𝑣∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
35		nfv 1921	. . . . . . . . . . 11 ⊢ Ⅎ𝑤 𝑢 ∈ 𝑥
36		nfopab1 5109	. . . . . . . . . . . 12 ⊢ Ⅎ𝑤{⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
37	36	nfel2 2918	. . . . . . . . . . 11 ⊢ Ⅎ𝑤⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
38	35, 37	nfan 1906	. . . . . . . . . 10 ⊢ Ⅎ𝑤(𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
39		nfv 1921	. . . . . . . . . 10 ⊢ Ⅎ𝑢(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
40		elequ1 2121	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑤 → (𝑢 ∈ 𝑥 ↔ 𝑤 ∈ 𝑥))
41		opeq1 4769	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑤 → ⟨𝑢, 𝑣⟩ = ⟨𝑤, 𝑣⟩)
42	41	eleq1d 2818	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑤 → (⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ↔ ⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}))
43	40, 42	anbi12d 634	. . . . . . . . . 10 ⊢ (𝑢 = 𝑤 → ((𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ (𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})))
44	38, 39, 43	cbvexv1 2345	. . . . . . . . 9 ⊢ (∃𝑢(𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}))
45		opeq2 4771	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑧 → ⟨𝑤, 𝑣⟩ = ⟨𝑤, 𝑧⟩)
46	45	eleq1d 2818	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑧 → (⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ↔ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}))
47	46	anbi2d 632	. . . . . . . . . 10 ⊢ (𝑣 = 𝑧 → ((𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ (𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})))
48	47	exbidv 1928	. . . . . . . . 9 ⊢ (𝑣 = 𝑧 → (∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})))
49	44, 48	syl5bb 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 2808	. . . . . . 7 ⊢ {𝑣 ∣ ∃𝑢(𝑢 ∈ 𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})} = {𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})}
51	28, 50	eqtri 2762	. . . . . 6 ⊢ ({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) = {𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})}
52	51	eleq1i 2824	. . . . 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 584	. 2 ⊢ ((∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) ∧ Fin = V) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
56	55	expcom 417	1 ⊢ (Fin = V → (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1540   = wceq 1542  ∃wex 1786   ∈ wcel 2114  ∃*wmo 2539  {cab 2717  Vcvv 3400  ⟨cop 4532  {copab 5102   “ cima 5538  Fun wfun 6343  Fincfn 8567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306  ax-un 7491

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-tr 5147  df-id 5439  df-eprel 5444  df-po 5452  df-so 5453  df-fr 5493  df-we 5495  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-ord 6185  df-on 6186  df-lim 6187  df-suc 6188  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-om 7612  df-1o 8143  df-en 8568  df-fin 8571

This theorem is referenced by: (None)