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Theorem 2nd2val 7314
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 5286 . . 3 (𝐴 ∈ (V × V) ↔ ∃𝑤𝑣 𝐴 = ⟨𝑤, 𝑣⟩)
2 fveq2 6304 . . . . . 6 (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘⟨𝑤, 𝑣⟩))
3 df-ov 6768 . . . . . . 7 (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}𝑣) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘⟨𝑤, 𝑣⟩)
4 vex 3307 . . . . . . . 8 𝑤 ∈ V
5 vex 3307 . . . . . . . 8 𝑣 ∈ V
6 simpr 479 . . . . . . . . 9 ((𝑥 = 𝑤𝑦 = 𝑣) → 𝑦 = 𝑣)
7 mpt2v 6867 . . . . . . . . . 10 (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝑦) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}
87eqcomi 2733 . . . . . . . . 9 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝑦)
96, 8, 5ovmpt2a 6908 . . . . . . . 8 ((𝑤 ∈ V ∧ 𝑣 ∈ V) → (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}𝑣) = 𝑣)
104, 5, 9mp2an 710 . . . . . . 7 (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}𝑣) = 𝑣
113, 10eqtr3i 2748 . . . . . 6 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘⟨𝑤, 𝑣⟩) = 𝑣
122, 11syl6eq 2774 . . . . 5 (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = 𝑣)
134, 5op2ndd 7296 . . . . 5 (𝐴 = ⟨𝑤, 𝑣⟩ → (2nd𝐴) = 𝑣)
1412, 13eqtr4d 2761 . . . 4 (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = (2nd𝐴))
1514exlimivv 1973 . . 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 3307 . . . . . . . . . 10 𝑥 ∈ V
18 vex 3307 . . . . . . . . . 10 𝑦 ∈ V
1917, 18pm3.2i 470 . . . . . . . . 9 (𝑥 ∈ V ∧ 𝑦 ∈ V)
20 ax6ev 2020 . . . . . . . . 9 𝑧 𝑧 = 𝑦
2119, 202th 254 . . . . . . . 8 ((𝑥 ∈ V ∧ 𝑦 ∈ V) ↔ ∃𝑧 𝑧 = 𝑦)
2221opabbii 4825 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 𝑧 = 𝑦}
23 df-xp 5224 . . . . . . 7 (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
24 dmoprab 6858 . . . . . . 7 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 𝑧 = 𝑦}
2522, 23, 243eqtr4ri 2757 . . . . . 6 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (V × V)
2625eleq2i 2795 . . . . 5 (𝐴 ∈ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} ↔ 𝐴 ∈ (V × V))
27 ndmfv 6331 . . . . 5 𝐴 ∈ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = ∅)
2826, 27sylnbir 320 . . . 4 𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = ∅)
29 rnsnn0 5711 . . . . . . . 8 (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅)
3029biimpri 218 . . . . . . 7 (ran {𝐴} ≠ ∅ → 𝐴 ∈ (V × V))
3130necon1bi 2924 . . . . . 6 𝐴 ∈ (V × V) → ran {𝐴} = ∅)
3231unieqd 4554 . . . . 5 𝐴 ∈ (V × V) → ran {𝐴} = ∅)
33 uni0 4573 . . . . 5 ∅ = ∅
3432, 33syl6eq 2774 . . . 4 𝐴 ∈ (V × V) → ran {𝐴} = ∅)
3528, 34eqtr4d 2761 . . 3 𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = ran {𝐴})
36 2ndval 7288 . . 3 (2nd𝐴) = ran {𝐴}
3735, 36syl6eqr 2776 . 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 383   = wceq 1596  wex 1817  wcel 2103  wne 2896  Vcvv 3304  c0 4023  {csn 4285  cop 4291   cuni 4544  {copab 4820   × cxp 5216  dom cdm 5218  ran crn 5219  cfv 6001  (class class class)co 6765  {coprab 6766  cmpt2 6767  2nd c2nd 7284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-iota 5964  df-fun 6003  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-2nd 7286
This theorem is referenced by: (None)
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