Theorem 2nd2val 8004

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 5751	. . 3 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑤∃𝑣 𝐴 = ⟨𝑤, 𝑣⟩)
2		fveq2 6892	. . . . . 6 ⊢ (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘⟨𝑤, 𝑣⟩))
3		df-ov 7412	. . . . . . 7 ⊢ (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}𝑣) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘⟨𝑤, 𝑣⟩)
4		simpr 486	. . . . . . . . 9 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → 𝑦 = 𝑣)
5		mpov 7520	. . . . . . . . . 10 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝑦) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}
6	5	eqcomi 2742	. . . . . . . . 9 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝑦)
7		vex 3479	. . . . . . . . 9 ⊢ 𝑣 ∈ V
8	4, 6, 7	ovmpoa 7563	. . . . . . . 8 ⊢ ((𝑤 ∈ V ∧ 𝑣 ∈ V) → (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}𝑣) = 𝑣)
9	8	el2v 3483	. . . . . . 7 ⊢ (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}𝑣) = 𝑣
10	3, 9	eqtr3i 2763	. . . . . 6 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘⟨𝑤, 𝑣⟩) = 𝑣
11	2, 10	eqtrdi 2789	. . . . 5 ⊢ (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = 𝑣)
12		vex 3479	. . . . . 6 ⊢ 𝑤 ∈ V
13	12, 7	op2ndd 7986	. . . . 5 ⊢ (𝐴 = ⟨𝑤, 𝑣⟩ → (2nd ‘𝐴) = 𝑣)
14	11, 13	eqtr4d 2776	. . . 4 ⊢ (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = (2nd ‘𝐴))
15	14	exlimivv 1936	. . 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 3479	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
18		vex 3479	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
19	17, 18	pm3.2i 472	. . . . . . . . 9 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
20		ax6ev 1974	. . . . . . . . 9 ⊢ ∃𝑧 𝑧 = 𝑦
21	19, 20	2th 264	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ↔ ∃𝑧 𝑧 = 𝑦)
22	21	opabbii 5216	. . . . . . 7 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 𝑧 = 𝑦}
23		df-xp 5683	. . . . . . 7 ⊢ (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
24		dmoprab 7510	. . . . . . 7 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 𝑧 = 𝑦}
25	22, 23, 24	3eqtr4ri 2772	. . . . . 6 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (V × V)
26	25	eleq2i 2826	. . . . 5 ⊢ (𝐴 ∈ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} ↔ 𝐴 ∈ (V × V))
27		ndmfv 6927	. . . . 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 6208	. . . . . . . 8 ⊢ (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅)
30	29	biimpri 227	. . . . . . 7 ⊢ (ran {𝐴} ≠ ∅ → 𝐴 ∈ (V × V))
31	30	necon1bi 2970	. . . . . 6 ⊢ (¬ 𝐴 ∈ (V × V) → ran {𝐴} = ∅)
32	31	unieqd 4923	. . . . 5 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ ran {𝐴} = ∪ ∅)
33		uni0 4940	. . . . 5 ⊢ ∪ ∅ = ∅
34	32, 33	eqtrdi 2789	. . . 4 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ ran {𝐴} = ∅)
35	28, 34	eqtr4d 2776	. . 3 ⊢ (¬ 𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = ∪ ran {𝐴})
36		2ndval 7978	. . 3 ⊢ (2nd ‘𝐴) = ∪ ran {𝐴}
37	35, 36	eqtr4di 2791	. 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 397   = wceq 1542  ∃wex 1782   ∈ wcel 2107   ≠ wne 2941  Vcvv 3475  ∅c0 4323  {csn 4629  ⟨cop 4635  ∪ cuni 4909  {copab 5211   × cxp 5675  dom cdm 5677  ran crn 5678  ‘cfv 6544  (class class class)co 7409  {coprab 7410   ∈ cmpo 7411  2^nd c2nd 7974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-2nd 7976

This theorem is referenced by: (None)