Theorem 1st2val 8013

Description: Value of an alternate definition of the 1^st function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 30-Dec-2014.)

Assertion

Ref	Expression
1st2val	⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = (1st ‘𝐴)

Distinct variable group:   𝑥,𝑦,𝑧

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)

Proof of Theorem 1st2val

Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elvv 5737	. . 3 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑤∃𝑣 𝐴 = ⟨𝑤, 𝑣⟩)
2		fveq2 6882	. . . . . 6 ⊢ (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘⟨𝑤, 𝑣⟩))
3		df-ov 7414	. . . . . . 7 ⊢ (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}𝑣) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘⟨𝑤, 𝑣⟩)
4		simpl 487	. . . . . . . . 9 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → 𝑥 = 𝑤)
5		mpov 7523	. . . . . . . . . 10 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝑥) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}
6	5	eqcomi 2778	. . . . . . . . 9 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝑥)
7		vex 3467	. . . . . . . . 9 ⊢ 𝑤 ∈ V
8	4, 6, 7	ovmpoa 7566	. . . . . . . 8 ⊢ ((𝑤 ∈ V ∧ 𝑣 ∈ V) → (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}𝑣) = 𝑤)
9	8	el2v 3470	. . . . . . 7 ⊢ (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}𝑣) = 𝑤
10	3, 9	eqtr3i 2794	. . . . . 6 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘⟨𝑤, 𝑣⟩) = 𝑤
11	2, 10	eqtrdi 2820	. . . . 5 ⊢ (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = 𝑤)
12		vex 3467	. . . . . 6 ⊢ 𝑣 ∈ V
13	7, 12	op1std 7995	. . . . 5 ⊢ (𝐴 = ⟨𝑤, 𝑣⟩ → (1st ‘𝐴) = 𝑤)
14	11, 13	eqtr4d 2807	. . . 4 ⊢ (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = (1st ‘𝐴))
15	14	exlimivv 1959	. . 3 ⊢ (∃𝑤∃𝑣 𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = (1st ‘𝐴))
16	1, 15	sylbi 220	. 2 ⊢ (𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = (1st ‘𝐴))
17		vex 3467	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
18		vex 3467	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
19	17, 18	pm3.2i 475	. . . . . . . . 9 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
20		ax6ev 1996	. . . . . . . . 9 ⊢ ∃𝑧 𝑧 = 𝑥
21	19, 20	2th 267	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ↔ ∃𝑧 𝑧 = 𝑥)
22	21	opabbii 5182	. . . . . . 7 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 𝑧 = 𝑥}
23		df-xp 5668	. . . . . . 7 ⊢ (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
24		dmoprab 7514	. . . . . . 7 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 𝑧 = 𝑥}
25	22, 23, 24	3eqtr4ri 2803	. . . . . 6 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (V × V)
26	25	eleq2i 2861	. . . . 5 ⊢ (𝐴 ∈ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} ↔ 𝐴 ∈ (V × V))
27		ndmfv 6914	. . . . 5 ⊢ (¬ 𝐴 ∈ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = ∅)
28	26, 27	sylnbir 334	. . . 4 ⊢ (¬ 𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = ∅)
29		dmsnn0 6209	. . . . . . . 8 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
30	29	biimpri 231	. . . . . . 7 ⊢ (dom {𝐴} ≠ ∅ → 𝐴 ∈ (V × V))
31	30	necon1bi 2992	. . . . . 6 ⊢ (¬ 𝐴 ∈ (V × V) → dom {𝐴} = ∅)
32	31	unieqd 4889	. . . . 5 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ dom {𝐴} = ∪ ∅)
33		uni0 4905	. . . . 5 ⊢ ∪ ∅ = ∅
34	32, 33	eqtrdi 2820	. . . 4 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ dom {𝐴} = ∅)
35	28, 34	eqtr4d 2807	. . 3 ⊢ (¬ 𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = ∪ dom {𝐴})
36		1stval 7987	. . 3 ⊢ (1st ‘𝐴) = ∪ dom {𝐴}
37	35, 36	eqtr4di 2822	. 2 ⊢ (¬ 𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = (1st ‘𝐴))
38	16, 37	pm2.61i 184	1 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = (1st ‘𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149   ≠ wne 2964  Vcvv 3463  ∅c0 4294  {csn 4594  ⟨cop 4600  ∪ cuni 4876  {copab 5177   × cxp 5660  dom cdm 5662  ‘cfv 6537  (class class class)co 7411  {coprab 7412   ∈ cmpo 7413  1^st c1st 7983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7985

This theorem is referenced by: (None)