Theorem 2nd2val 8043

Description: Value of an alternate definition of the 2^nd function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 30-Dec-2014.)

Assertion

Ref	Expression
2nd2val	⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = (2nd ‘𝐴)

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.

Step	Hyp	Ref	Expression
1		elvv 5760	. . 3 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑤∃𝑣 𝐴 = ⟨𝑤, 𝑣⟩)
2		fveq2 6906	. . . . . 6 ⊢ (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘⟨𝑤, 𝑣⟩))
3		df-ov 7434	. . . . . . 7 ⊢ (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}𝑣) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘⟨𝑤, 𝑣⟩)
4		simpr 484	. . . . . . . . 9 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → 𝑦 = 𝑣)
5		mpov 7545	. . . . . . . . . 10 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝑦) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}
6	5	eqcomi 2746	. . . . . . . . 9 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝑦)
7		vex 3484	. . . . . . . . 9 ⊢ 𝑣 ∈ V
8	4, 6, 7	ovmpoa 7588	. . . . . . . 8 ⊢ ((𝑤 ∈ V ∧ 𝑣 ∈ V) → (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}𝑣) = 𝑣)
9	8	el2v 3487	. . . . . . 7 ⊢ (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}𝑣) = 𝑣
10	3, 9	eqtr3i 2767	. . . . . 6 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘⟨𝑤, 𝑣⟩) = 𝑣
11	2, 10	eqtrdi 2793	. . . . 5 ⊢ (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = 𝑣)
12		vex 3484	. . . . . 6 ⊢ 𝑤 ∈ V
13	12, 7	op2ndd 8025	. . . . 5 ⊢ (𝐴 = ⟨𝑤, 𝑣⟩ → (2nd ‘𝐴) = 𝑣)
14	11, 13	eqtr4d 2780	. . . 4 ⊢ (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = (2nd ‘𝐴))
15	14	exlimivv 1932	. . 3 ⊢ (∃𝑤∃𝑣 𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = (2nd ‘𝐴))
16	1, 15	sylbi 217	. 2 ⊢ (𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = (2nd ‘𝐴))
17		vex 3484	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
18		vex 3484	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
19	17, 18	pm3.2i 470	. . . . . . . . 9 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
20		ax6ev 1969	. . . . . . . . 9 ⊢ ∃𝑧 𝑧 = 𝑦
21	19, 20	2th 264	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ↔ ∃𝑧 𝑧 = 𝑦)
22	21	opabbii 5210	. . . . . . 7 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 𝑧 = 𝑦}
23		df-xp 5691	. . . . . . 7 ⊢ (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
24		dmoprab 7536	. . . . . . 7 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 𝑧 = 𝑦}
25	22, 23, 24	3eqtr4ri 2776	. . . . . 6 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (V × V)
26	25	eleq2i 2833	. . . . 5 ⊢ (𝐴 ∈ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} ↔ 𝐴 ∈ (V × V))
27		ndmfv 6941	. . . . 5 ⊢ (¬ 𝐴 ∈ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = ∅)
28	26, 27	sylnbir 331	. . . 4 ⊢ (¬ 𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = ∅)
29		rnsnn0 6228	. . . . . . . 8 ⊢ (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅)
30	29	biimpri 228	. . . . . . 7 ⊢ (ran {𝐴} ≠ ∅ → 𝐴 ∈ (V × V))
31	30	necon1bi 2969	. . . . . 6 ⊢ (¬ 𝐴 ∈ (V × V) → ran {𝐴} = ∅)
32	31	unieqd 4920	. . . . 5 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ ran {𝐴} = ∪ ∅)
33		uni0 4935	. . . . 5 ⊢ ∪ ∅ = ∅
34	32, 33	eqtrdi 2793	. . . 4 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ ran {𝐴} = ∅)
35	28, 34	eqtr4d 2780	. . 3 ⊢ (¬ 𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = ∪ ran {𝐴})
36		2ndval 8017	. . 3 ⊢ (2nd ‘𝐴) = ∪ ran {𝐴}
37	35, 36	eqtr4di 2795	. 2 ⊢ (¬ 𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = (2nd ‘𝐴))
38	16, 37	pm2.61i 182	1 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = (2nd ‘𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108   ≠ wne 2940  Vcvv 3480  ∅c0 4333  {csn 4626  ⟨cop 4632  ∪ cuni 4907  {copab 5205   × cxp 5683  dom cdm 5685  ran crn 5686  ‘cfv 6561  (class class class)co 7431  {coprab 7432   ∈ cmpo 7433  2^nd c2nd 8013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8015

This theorem is referenced by: (None)