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Theorem 1st2val 7949
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 5706 . . 3 (𝐴 ∈ (V × V) ↔ ∃𝑤𝑣 𝐴 = ⟨𝑤, 𝑣⟩)
2 fveq2 6842 . . . . . 6 (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘⟨𝑤, 𝑣⟩))
3 df-ov 7360 . . . . . . 7 (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}𝑣) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘⟨𝑤, 𝑣⟩)
4 simpl 483 . . . . . . . . 9 ((𝑥 = 𝑤𝑦 = 𝑣) → 𝑥 = 𝑤)
5 mpov 7468 . . . . . . . . . 10 (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝑥) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}
65eqcomi 2745 . . . . . . . . 9 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝑥)
7 vex 3449 . . . . . . . . 9 𝑤 ∈ V
84, 6, 7ovmpoa 7510 . . . . . . . 8 ((𝑤 ∈ V ∧ 𝑣 ∈ V) → (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}𝑣) = 𝑤)
98el2v 3453 . . . . . . 7 (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}𝑣) = 𝑤
103, 9eqtr3i 2766 . . . . . 6 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘⟨𝑤, 𝑣⟩) = 𝑤
112, 10eqtrdi 2792 . . . . 5 (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = 𝑤)
12 vex 3449 . . . . . 6 𝑣 ∈ V
137, 12op1std 7931 . . . . 5 (𝐴 = ⟨𝑤, 𝑣⟩ → (1st𝐴) = 𝑤)
1411, 13eqtr4d 2779 . . . 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 3449 . . . . . . . . . 10 𝑥 ∈ V
18 vex 3449 . . . . . . . . . 10 𝑦 ∈ V
1917, 18pm3.2i 471 . . . . . . . . 9 (𝑥 ∈ V ∧ 𝑦 ∈ V)
20 ax6ev 1973 . . . . . . . . 9 𝑧 𝑧 = 𝑥
2119, 202th 263 . . . . . . . 8 ((𝑥 ∈ V ∧ 𝑦 ∈ V) ↔ ∃𝑧 𝑧 = 𝑥)
2221opabbii 5172 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 𝑧 = 𝑥}
23 df-xp 5639 . . . . . . 7 (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
24 dmoprab 7458 . . . . . . 7 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 𝑧 = 𝑥}
2522, 23, 243eqtr4ri 2775 . . . . . 6 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (V × V)
2625eleq2i 2829 . . . . 5 (𝐴 ∈ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} ↔ 𝐴 ∈ (V × V))
27 ndmfv 6877 . . . . 5 𝐴 ∈ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = ∅)
2826, 27sylnbir 330 . . . 4 𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = ∅)
29 dmsnn0 6159 . . . . . . . 8 (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
3029biimpri 227 . . . . . . 7 (dom {𝐴} ≠ ∅ → 𝐴 ∈ (V × V))
3130necon1bi 2972 . . . . . 6 𝐴 ∈ (V × V) → dom {𝐴} = ∅)
3231unieqd 4879 . . . . 5 𝐴 ∈ (V × V) → dom {𝐴} = ∅)
33 uni0 4896 . . . . 5 ∅ = ∅
3432, 33eqtrdi 2792 . . . 4 𝐴 ∈ (V × V) → dom {𝐴} = ∅)
3528, 34eqtr4d 2779 . . 3 𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = dom {𝐴})
36 1stval 7923 . . 3 (1st𝐴) = dom {𝐴}
3735, 36eqtr4di 2794 . 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 396   = wceq 1541  wex 1781  wcel 2106  wne 2943  Vcvv 3445  c0 4282  {csn 4586  cop 4592   cuni 4865  {copab 5167   × cxp 5631  dom cdm 5633  cfv 6496  (class class class)co 7357  {coprab 7358  cmpo 7359  1st c1st 7919
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 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921
This theorem is referenced by: (None)
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