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Theorem 1st2val 7357
Description: Value of an alternate definition of the 1st function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 30-Dec-2014.)
Assertion
Ref Expression
1st2val ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = (1st𝐴)
Distinct variable group:   𝑥,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)

Proof of Theorem 1st2val
Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elvv 5330 . . 3 (𝐴 ∈ (V × V) ↔ ∃𝑤𝑣 𝐴 = ⟨𝑤, 𝑣⟩)
2 fveq2 6348 . . . . . 6 (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘⟨𝑤, 𝑣⟩))
3 df-ov 6812 . . . . . . 7 (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}𝑣) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘⟨𝑤, 𝑣⟩)
4 vex 3339 . . . . . . . 8 𝑤 ∈ V
5 vex 3339 . . . . . . . 8 𝑣 ∈ V
6 simpl 474 . . . . . . . . 9 ((𝑥 = 𝑤𝑦 = 𝑣) → 𝑥 = 𝑤)
7 mpt2v 6911 . . . . . . . . . 10 (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝑥) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}
87eqcomi 2765 . . . . . . . . 9 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝑥)
96, 8, 4ovmpt2a 6952 . . . . . . . 8 ((𝑤 ∈ V ∧ 𝑣 ∈ V) → (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}𝑣) = 𝑤)
104, 5, 9mp2an 710 . . . . . . 7 (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}𝑣) = 𝑤
113, 10eqtr3i 2780 . . . . . 6 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘⟨𝑤, 𝑣⟩) = 𝑤
122, 11syl6eq 2806 . . . . 5 (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = 𝑤)
134, 5op1std 7339 . . . . 5 (𝐴 = ⟨𝑤, 𝑣⟩ → (1st𝐴) = 𝑤)
1412, 13eqtr4d 2793 . . . 4 (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = (1st𝐴))
1514exlimivv 2005 . . 3 (∃𝑤𝑣 𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = (1st𝐴))
161, 15sylbi 207 . 2 (𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = (1st𝐴))
17 vex 3339 . . . . . . . . . 10 𝑥 ∈ V
18 vex 3339 . . . . . . . . . 10 𝑦 ∈ V
1917, 18pm3.2i 470 . . . . . . . . 9 (𝑥 ∈ V ∧ 𝑦 ∈ V)
20 ax6ev 2052 . . . . . . . . 9 𝑧 𝑧 = 𝑥
2119, 202th 254 . . . . . . . 8 ((𝑥 ∈ V ∧ 𝑦 ∈ V) ↔ ∃𝑧 𝑧 = 𝑥)
2221opabbii 4865 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 𝑧 = 𝑥}
23 df-xp 5268 . . . . . . 7 (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
24 dmoprab 6902 . . . . . . 7 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 𝑧 = 𝑥}
2522, 23, 243eqtr4ri 2789 . . . . . 6 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (V × V)
2625eleq2i 2827 . . . . 5 (𝐴 ∈ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} ↔ 𝐴 ∈ (V × V))
27 ndmfv 6375 . . . . 5 𝐴 ∈ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = ∅)
2826, 27sylnbir 320 . . . 4 𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = ∅)
29 dmsnn0 5754 . . . . . . . 8 (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
3029biimpri 218 . . . . . . 7 (dom {𝐴} ≠ ∅ → 𝐴 ∈ (V × V))
3130necon1bi 2956 . . . . . 6 𝐴 ∈ (V × V) → dom {𝐴} = ∅)
3231unieqd 4594 . . . . 5 𝐴 ∈ (V × V) → dom {𝐴} = ∅)
33 uni0 4613 . . . . 5 ∅ = ∅
3432, 33syl6eq 2806 . . . 4 𝐴 ∈ (V × V) → dom {𝐴} = ∅)
3528, 34eqtr4d 2793 . . 3 𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = dom {𝐴})
36 1stval 7331 . . 3 (1st𝐴) = dom {𝐴}
3735, 36syl6eqr 2808 . 2 𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = (1st𝐴))
3816, 37pm2.61i 176 1 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = (1st𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383   = wceq 1628  wex 1849  wcel 2135  wne 2928  Vcvv 3336  c0 4054  {csn 4317  cop 4323   cuni 4584  {copab 4860   × cxp 5260  dom cdm 5262  cfv 6045  (class class class)co 6809  {coprab 6810  cmpt2 6811  1st c1st 7327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-sbc 3573  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-br 4801  df-opab 4861  df-mpt 4878  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-iota 6008  df-fun 6047  df-fv 6053  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-1st 7329
This theorem is referenced by: (None)
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