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Theorem 2nd2val 7140
Description: Value of an alternate definition of the 2nd function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 30-Dec-2014.)
Assertion
Ref Expression
2nd2val ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = (2nd𝐴)
Distinct variable group:   𝑥,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)

Proof of Theorem 2nd2val
Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elvv 5138 . . 3 (𝐴 ∈ (V × V) ↔ ∃𝑤𝑣 𝐴 = ⟨𝑤, 𝑣⟩)
2 fveq2 6148 . . . . . 6 (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘⟨𝑤, 𝑣⟩))
3 df-ov 6607 . . . . . . 7 (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}𝑣) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘⟨𝑤, 𝑣⟩)
4 vex 3189 . . . . . . . 8 𝑤 ∈ V
5 vex 3189 . . . . . . . 8 𝑣 ∈ V
6 simpr 477 . . . . . . . . 9 ((𝑥 = 𝑤𝑦 = 𝑣) → 𝑦 = 𝑣)
7 mpt2v 6703 . . . . . . . . . 10 (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝑦) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}
87eqcomi 2630 . . . . . . . . 9 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝑦)
96, 8, 5ovmpt2a 6744 . . . . . . . 8 ((𝑤 ∈ V ∧ 𝑣 ∈ V) → (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}𝑣) = 𝑣)
104, 5, 9mp2an 707 . . . . . . 7 (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}𝑣) = 𝑣
113, 10eqtr3i 2645 . . . . . 6 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘⟨𝑤, 𝑣⟩) = 𝑣
122, 11syl6eq 2671 . . . . 5 (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = 𝑣)
134, 5op2ndd 7124 . . . . 5 (𝐴 = ⟨𝑤, 𝑣⟩ → (2nd𝐴) = 𝑣)
1412, 13eqtr4d 2658 . . . 4 (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = (2nd𝐴))
1514exlimivv 1857 . . 3 (∃𝑤𝑣 𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = (2nd𝐴))
161, 15sylbi 207 . 2 (𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = (2nd𝐴))
17 vex 3189 . . . . . . . . . 10 𝑥 ∈ V
18 vex 3189 . . . . . . . . . 10 𝑦 ∈ V
1917, 18pm3.2i 471 . . . . . . . . 9 (𝑥 ∈ V ∧ 𝑦 ∈ V)
20 ax6ev 1887 . . . . . . . . 9 𝑧 𝑧 = 𝑦
2119, 202th 254 . . . . . . . 8 ((𝑥 ∈ V ∧ 𝑦 ∈ V) ↔ ∃𝑧 𝑧 = 𝑦)
2221opabbii 4679 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 𝑧 = 𝑦}
23 df-xp 5080 . . . . . . 7 (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
24 dmoprab 6694 . . . . . . 7 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 𝑧 = 𝑦}
2522, 23, 243eqtr4ri 2654 . . . . . 6 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (V × V)
2625eleq2i 2690 . . . . 5 (𝐴 ∈ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} ↔ 𝐴 ∈ (V × V))
27 ndmfv 6175 . . . . 5 𝐴 ∈ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = ∅)
2826, 27sylnbir 321 . . . 4 𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = ∅)
29 rnsnn0 5560 . . . . . . . 8 (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅)
3029biimpri 218 . . . . . . 7 (ran {𝐴} ≠ ∅ → 𝐴 ∈ (V × V))
3130necon1bi 2818 . . . . . 6 𝐴 ∈ (V × V) → ran {𝐴} = ∅)
3231unieqd 4412 . . . . 5 𝐴 ∈ (V × V) → ran {𝐴} = ∅)
33 uni0 4431 . . . . 5 ∅ = ∅
3432, 33syl6eq 2671 . . . 4 𝐴 ∈ (V × V) → ran {𝐴} = ∅)
3528, 34eqtr4d 2658 . . 3 𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = ran {𝐴})
36 2ndval 7116 . . 3 (2nd𝐴) = ran {𝐴}
3735, 36syl6eqr 2673 . 2 𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = (2nd𝐴))
3816, 37pm2.61i 176 1 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = (2nd𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  Vcvv 3186  c0 3891  {csn 4148  cop 4154   cuni 4402  {copab 4672   × cxp 5072  dom cdm 5074  ran crn 5075  cfv 5847  (class class class)co 6604  {coprab 6605  cmpt2 6606  2nd c2nd 7112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-2nd 7114
This theorem is referenced by: (None)
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