Theorem 2nd2val 8016

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 5746	. . 3 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑤∃𝑣 𝐴 = ⟨𝑤, 𝑣⟩)
2		fveq2 6891	. . . . . 6 ⊢ (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘⟨𝑤, 𝑣⟩))
3		df-ov 7417	. . . . . . 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 7526	. . . . . . . . . 10 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝑦) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}
6	5	eqcomi 2736	. . . . . . . . 9 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝑦)
7		vex 3473	. . . . . . . . 9 ⊢ 𝑣 ∈ V
8	4, 6, 7	ovmpoa 7570	. . . . . . . 8 ⊢ ((𝑤 ∈ V ∧ 𝑣 ∈ V) → (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}𝑣) = 𝑣)
9	8	el2v 3477	. . . . . . 7 ⊢ (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}𝑣) = 𝑣
10	3, 9	eqtr3i 2757	. . . . . 6 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘⟨𝑤, 𝑣⟩) = 𝑣
11	2, 10	eqtrdi 2783	. . . . 5 ⊢ (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = 𝑣)
12		vex 3473	. . . . . 6 ⊢ 𝑤 ∈ V
13	12, 7	op2ndd 7998	. . . . 5 ⊢ (𝐴 = ⟨𝑤, 𝑣⟩ → (2nd ‘𝐴) = 𝑣)
14	11, 13	eqtr4d 2770	. . . 4 ⊢ (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = (2nd ‘𝐴))
15	14	exlimivv 1928	. . 3 ⊢ (∃𝑤∃𝑣 𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = (2nd ‘𝐴))
16	1, 15	sylbi 216	. 2 ⊢ (𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = (2nd ‘𝐴))
17		vex 3473	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
18		vex 3473	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
19	17, 18	pm3.2i 470	. . . . . . . . 9 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
20		ax6ev 1966	. . . . . . . . 9 ⊢ ∃𝑧 𝑧 = 𝑦
21	19, 20	2th 264	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ↔ ∃𝑧 𝑧 = 𝑦)
22	21	opabbii 5209	. . . . . . 7 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 𝑧 = 𝑦}
23		df-xp 5678	. . . . . . 7 ⊢ (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
24		dmoprab 7516	. . . . . . 7 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 𝑧 = 𝑦}
25	22, 23, 24	3eqtr4ri 2766	. . . . . 6 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (V × V)
26	25	eleq2i 2820	. . . . 5 ⊢ (𝐴 ∈ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} ↔ 𝐴 ∈ (V × V))
27		ndmfv 6926	. . . . 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 6206	. . . . . . . 8 ⊢ (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅)
30	29	biimpri 227	. . . . . . 7 ⊢ (ran {𝐴} ≠ ∅ → 𝐴 ∈ (V × V))
31	30	necon1bi 2964	. . . . . 6 ⊢ (¬ 𝐴 ∈ (V × V) → ran {𝐴} = ∅)
32	31	unieqd 4916	. . . . 5 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ ran {𝐴} = ∪ ∅)
33		uni0 4933	. . . . 5 ⊢ ∪ ∅ = ∅
34	32, 33	eqtrdi 2783	. . . 4 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ ran {𝐴} = ∅)
35	28, 34	eqtr4d 2770	. . 3 ⊢ (¬ 𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = ∪ ran {𝐴})
36		2ndval 7990	. . 3 ⊢ (2nd ‘𝐴) = ∪ ran {𝐴}
37	35, 36	eqtr4di 2785	. 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 1534  ∃wex 1774   ∈ wcel 2099   ≠ wne 2935  Vcvv 3469  ∅c0 4318  {csn 4624  ⟨cop 4630  ∪ cuni 4903  {copab 5204   × cxp 5670  dom cdm 5672  ran crn 5673  ‘cfv 6542  (class class class)co 7414  {coprab 7415   ∈ cmpo 7416  2^nd c2nd 7986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-2nd 7988

This theorem is referenced by: (None)