Theorem 2nd2val 8030

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 5725	. . 3 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑤∃𝑣 𝐴 = ⟨𝑤, 𝑣⟩)
2		fveq2 6885	. . . . . 6 ⊢ (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘⟨𝑤, 𝑣⟩))
3		df-ov 7423	. . . . . . 7 ⊢ (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}𝑣) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘⟨𝑤, 𝑣⟩)
4		simpr 490	. . . . . . . . 9 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → 𝑦 = 𝑣)
5		mpov 7532	. . . . . . . . . 10 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝑦) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}
6	5	eqcomi 2769	. . . . . . . . 9 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝑦)
7		vex 3454	. . . . . . . . 9 ⊢ 𝑣 ∈ V
8	4, 6, 7	ovmpoa 7575	. . . . . . . 8 ⊢ ((𝑤 ∈ V ∧ 𝑣 ∈ V) → (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}𝑣) = 𝑣)
9	8	el2v 3457	. . . . . . 7 ⊢ (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}𝑣) = 𝑣
10	3, 9	eqtr3i 2785	. . . . . 6 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘⟨𝑤, 𝑣⟩) = 𝑣
11	2, 10	eqtrdi 2811	. . . . 5 ⊢ (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = 𝑣)
12		vex 3454	. . . . . 6 ⊢ 𝑤 ∈ V
13	12, 7	op2ndd 8012	. . . . 5 ⊢ (𝐴 = ⟨𝑤, 𝑣⟩ → (2nd ‘𝐴) = 𝑣)
14	11, 13	eqtr4d 2798	. . . 4 ⊢ (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = (2nd ‘𝐴))
15	14	exlimivv 1964	. . 3 ⊢ (∃𝑤∃𝑣 𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = (2nd ‘𝐴))
16	1, 15	sylbi 220	. 2 ⊢ (𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = (2nd ‘𝐴))
17		vex 3454	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
18		vex 3454	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
19	17, 18	pm3.2i 476	. . . . . . . . 9 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
20		ax6ev 2001	. . . . . . . . 9 ⊢ ∃𝑧 𝑧 = 𝑦
21	19, 20	2th 267	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ↔ ∃𝑧 𝑧 = 𝑦)
22	21	opabbii 5171	. . . . . . 7 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 𝑧 = 𝑦}
23		df-xp 5656	. . . . . . 7 ⊢ (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
24		dmoprab 7523	. . . . . . 7 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 𝑧 = 𝑦}
25	22, 23, 24	3eqtr4ri 2794	. . . . . 6 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (V × V)
26	25	eleq2i 2852	. . . . 5 ⊢ (𝐴 ∈ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} ↔ 𝐴 ∈ (V × V))
27		ndmfv 6917	. . . . 5 ⊢ (¬ 𝐴 ∈ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = ∅)
28	26, 27	sylnbir 334	. . . 4 ⊢ (¬ 𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = ∅)
29		rnsnn0 6209	. . . . . . . 8 ⊢ (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅)
30	29	biimpri 231	. . . . . . 7 ⊢ (ran {𝐴} ≠ ∅ → 𝐴 ∈ (V × V))
31	30	necon1bi 2983	. . . . . 6 ⊢ (¬ 𝐴 ∈ (V × V) → ran {𝐴} = ∅)
32	31	unieqd 4879	. . . . 5 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ ran {𝐴} = ∪ ∅)
33		uni0 4895	. . . . 5 ⊢ ∪ ∅ = ∅
34	32, 33	eqtrdi 2811	. . . 4 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ ran {𝐴} = ∅)
35	28, 34	eqtr4d 2798	. . 3 ⊢ (¬ 𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = ∪ ran {𝐴})
36		2ndval 8004	. . 3 ⊢ (2nd ‘𝐴) = ∪ ran {𝐴}
37	35, 36	eqtr4di 2813	. 2 ⊢ (¬ 𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = (2nd ‘𝐴))
38	16, 37	pm2.61i 184	1 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = (2nd ‘𝐴)

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144   ≠ wne 2955  Vcvv 3450  ∅c0 4278  {csn 4583  ⟨cop 4589  ∪ cuni 4866  {copab 5166   × cxp 5648  dom cdm 5650  ran crn 5651  ‘cfv 6538  (class class class)co 7420  {coprab 7421   ∈ cmpo 7422  2^nd c2nd 8000

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-2nd 8002

This theorem is used by: (None)