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Theorem 1st2val 7932
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 5697 . . 3 (𝐴 ∈ (V × V) ↔ ∃𝑤𝑣 𝐴 = ⟨𝑤, 𝑣⟩)
2 fveq2 6830 . . . . . 6 (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘⟨𝑤, 𝑣⟩))
3 df-ov 7345 . . . . . . 7 (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}𝑣) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘⟨𝑤, 𝑣⟩)
4 simpl 484 . . . . . . . . 9 ((𝑥 = 𝑤𝑦 = 𝑣) → 𝑥 = 𝑤)
5 mpov 7453 . . . . . . . . . 10 (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝑥) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}
65eqcomi 2746 . . . . . . . . 9 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝑥)
7 vex 3446 . . . . . . . . 9 𝑤 ∈ V
84, 6, 7ovmpoa 7495 . . . . . . . 8 ((𝑤 ∈ V ∧ 𝑣 ∈ V) → (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}𝑣) = 𝑤)
98el2v 3450 . . . . . . 7 (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}𝑣) = 𝑤
103, 9eqtr3i 2767 . . . . . 6 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘⟨𝑤, 𝑣⟩) = 𝑤
112, 10eqtrdi 2793 . . . . 5 (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = 𝑤)
12 vex 3446 . . . . . 6 𝑣 ∈ V
137, 12op1std 7914 . . . . 5 (𝐴 = ⟨𝑤, 𝑣⟩ → (1st𝐴) = 𝑤)
1411, 13eqtr4d 2780 . . . 4 (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = (1st𝐴))
1514exlimivv 1935 . . 3 (∃𝑤𝑣 𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = (1st𝐴))
161, 15sylbi 216 . 2 (𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = (1st𝐴))
17 vex 3446 . . . . . . . . . 10 𝑥 ∈ V
18 vex 3446 . . . . . . . . . 10 𝑦 ∈ V
1917, 18pm3.2i 472 . . . . . . . . 9 (𝑥 ∈ V ∧ 𝑦 ∈ V)
20 ax6ev 1973 . . . . . . . . 9 𝑧 𝑧 = 𝑥
2119, 202th 264 . . . . . . . 8 ((𝑥 ∈ V ∧ 𝑦 ∈ V) ↔ ∃𝑧 𝑧 = 𝑥)
2221opabbii 5164 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 𝑧 = 𝑥}
23 df-xp 5631 . . . . . . 7 (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
24 dmoprab 7443 . . . . . . 7 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 𝑧 = 𝑥}
2522, 23, 243eqtr4ri 2776 . . . . . 6 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (V × V)
2625eleq2i 2829 . . . . 5 (𝐴 ∈ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} ↔ 𝐴 ∈ (V × V))
27 ndmfv 6865 . . . . 5 𝐴 ∈ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = ∅)
2826, 27sylnbir 331 . . . 4 𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = ∅)
29 dmsnn0 6150 . . . . . . . 8 (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
3029biimpri 227 . . . . . . 7 (dom {𝐴} ≠ ∅ → 𝐴 ∈ (V × V))
3130necon1bi 2970 . . . . . 6 𝐴 ∈ (V × V) → dom {𝐴} = ∅)
3231unieqd 4871 . . . . 5 𝐴 ∈ (V × V) → dom {𝐴} = ∅)
33 uni0 4888 . . . . 5 ∅ = ∅
3432, 33eqtrdi 2793 . . . 4 𝐴 ∈ (V × V) → dom {𝐴} = ∅)
3528, 34eqtr4d 2780 . . 3 𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = dom {𝐴})
36 1stval 7906 . . 3 (1st𝐴) = dom {𝐴}
3735, 36eqtr4di 2795 . 2 𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = (1st𝐴))
3816, 37pm2.61i 182 1 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = (1st𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 397   = wceq 1541  wex 1781  wcel 2106  wne 2941  Vcvv 3442  c0 4274  {csn 4578  cop 4584   cuni 4857  {copab 5159   × cxp 5623  dom cdm 5625  cfv 6484  (class class class)co 7342  {coprab 7343  cmpo 7344  1st c1st 7902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5248  ax-nul 5255  ax-pr 5377  ax-un 7655
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3444  df-sbc 3732  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4275  df-if 4479  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4858  df-br 5098  df-opab 5160  df-mpt 5181  df-id 5523  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6436  df-fun 6486  df-fv 6492  df-ov 7345  df-oprab 7346  df-mpo 7347  df-1st 7904
This theorem is referenced by: (None)
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