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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 → (∀𝑤𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀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.
StepHypRef Expression
1 funopab 6384 . . . 4 (Fun {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ↔ ∀𝑤∃*𝑧𝑦𝜑)
2 nfa1 2156 . . . . . 6 𝑦𝑦𝜑
32mof 2564 . . . . 5 (∃*𝑧𝑦𝜑 ↔ ∃𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦))
43albii 1826 . . . 4 (∀𝑤∃*𝑧𝑦𝜑 ↔ ∀𝑤𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦))
51, 4bitr2i 279 . . 3 (∀𝑤𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) ↔ Fun {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
6 vex 3404 . . . . . . 7 𝑥 ∈ V
7 eleq2w2 2735 . . . . . . 7 (Fin = V → (𝑥 ∈ Fin ↔ 𝑥 ∈ V))
86, 7mpbiri 261 . . . . . 6 (Fin = V → 𝑥 ∈ Fin)
9 imafi 8785 . . . . . 6 ((Fun {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ∧ 𝑥 ∈ Fin) → ({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) ∈ Fin)
108, 9sylan2 596 . . . . 5 ((Fun {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ∧ Fin = V) → ({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) ∈ Fin)
1110elexd 3420 . . . 4 ((Fun {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ∧ Fin = V) → ({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) ∈ V)
12 nfv 1921 . . . . . . . . . 10 𝑦 𝑤𝑥
132nfopab 5108 . . . . . . . . . . 11 𝑦{⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
1413nfel2 2918 . . . . . . . . . 10 𝑦𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
1512, 14nfan 1906 . . . . . . . . 9 𝑦(𝑤𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
1615nfex 2327 . . . . . . . 8 𝑦𝑤(𝑤𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
1716nfab 2906 . . . . . . 7 𝑦{𝑧 ∣ ∃𝑤(𝑤𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})}
1817issetf 3414 . . . . . 6 ({𝑧 ∣ ∃𝑤(𝑤𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})} ∈ V ↔ ∃𝑦 𝑦 = {𝑧 ∣ ∃𝑤(𝑤𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})})
19 abeq2 2865 . . . . . . 7 (𝑦 = {𝑧 ∣ ∃𝑤(𝑤𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})} ↔ ∀𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})))
2019exbii 1854 . . . . . 6 (∃𝑦 𝑦 = {𝑧 ∣ ∃𝑤(𝑤𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})} ↔ ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})))
21 opabidw 5390 . . . . . . . . . . 11 (⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ↔ ∀𝑦𝜑)
2221anbi2i 626 . . . . . . . . . 10 ((𝑤𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ (𝑤𝑥 ∧ ∀𝑦𝜑))
2322exbii 1854 . . . . . . . . 9 (∃𝑤(𝑤𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑))
2423bibi2i 341 . . . . . . . 8 ((𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})) ↔ (𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑)))
2524albii 1826 . . . . . . 7 (∀𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})) ↔ ∀𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑)))
2625exbii 1854 . . . . . 6 (∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})) ↔ ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑)))
2718, 20, 263bitrri 301 . . . . 5 (∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑)) ↔ {𝑧 ∣ ∃𝑤(𝑤𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})} ∈ V)
28 dfima3 5916 . . . . . . 7 ({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) = {𝑣 ∣ ∃𝑢(𝑢𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})}
29 nfv 1921 . . . . . . . . . 10 𝑧 𝑢𝑥
30 nfopab2 5110 . . . . . . . . . . 11 𝑧{⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
3130nfel2 2918 . . . . . . . . . 10 𝑧𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
3229, 31nfan 1906 . . . . . . . . 9 𝑧(𝑢𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
3332nfex 2327 . . . . . . . 8 𝑧𝑢(𝑢𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
34 nfv 1921 . . . . . . . 8 𝑣𝑤(𝑤𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
35 nfv 1921 . . . . . . . . . . 11 𝑤 𝑢𝑥
36 nfopab1 5109 . . . . . . . . . . . 12 𝑤{⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
3736nfel2 2918 . . . . . . . . . . 11 𝑤𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}
3835, 37nfan 1906 . . . . . . . . . 10 𝑤(𝑢𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
39 nfv 1921 . . . . . . . . . 10 𝑢(𝑤𝑥 ∧ ⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})
40 elequ1 2121 . . . . . . . . . . 11 (𝑢 = 𝑤 → (𝑢𝑥𝑤𝑥))
41 opeq1 4769 . . . . . . . . . . . 12 (𝑢 = 𝑤 → ⟨𝑢, 𝑣⟩ = ⟨𝑤, 𝑣⟩)
4241eleq1d 2818 . . . . . . . . . . 11 (𝑢 = 𝑤 → (⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ↔ ⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}))
4340, 42anbi12d 634 . . . . . . . . . 10 (𝑢 = 𝑤 → ((𝑢𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ (𝑤𝑥 ∧ ⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})))
4438, 39, 43cbvexv1 2345 . . . . . . . . 9 (∃𝑢(𝑢𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ ∃𝑤(𝑤𝑥 ∧ ⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}))
45 opeq2 4771 . . . . . . . . . . . 12 (𝑣 = 𝑧 → ⟨𝑤, 𝑣⟩ = ⟨𝑤, 𝑧⟩)
4645eleq1d 2818 . . . . . . . . . . 11 (𝑣 = 𝑧 → (⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ↔ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}))
4746anbi2d 632 . . . . . . . . . 10 (𝑣 = 𝑧 → ((𝑤𝑥 ∧ ⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ (𝑤𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})))
4847exbidv 1928 . . . . . . . . 9 (𝑣 = 𝑧 → (∃𝑤(𝑤𝑥 ∧ ⟨𝑤, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ ∃𝑤(𝑤𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})))
4944, 48syl5bb 286 . . . . . . . 8 (𝑣 = 𝑧 → (∃𝑢(𝑢𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑}) ↔ ∃𝑤(𝑤𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})))
5033, 34, 49cbvabw 2808 . . . . . . 7 {𝑣 ∣ ∃𝑢(𝑢𝑥 ∧ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})} = {𝑧 ∣ ∃𝑤(𝑤𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})}
5128, 50eqtri 2762 . . . . . 6 ({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) = {𝑧 ∣ ∃𝑤(𝑤𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})}
5251eleq1i 2824 . . . . 5 (({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) ∈ V ↔ {𝑧 ∣ ∃𝑤(𝑤𝑥 ∧ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑})} ∈ V)
5327, 52bitr4i 281 . . . 4 (∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑)) ↔ ({⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} “ 𝑥) ∈ V)
5411, 53sylibr 237 . . 3 ((Fun {⟨𝑤, 𝑧⟩ ∣ ∀𝑦𝜑} ∧ Fin = V) → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑)))
555, 54sylanb 584 . 2 ((∀𝑤𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) ∧ Fin = V) → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑)))
5655expcom 417 1 (Fin = V → (∀𝑤𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑))))
Colors of variables: wff setvar class
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)
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