Theorem 1st2val 8040

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 5762	. . 3 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑤∃𝑣 𝐴 = ⟨𝑤, 𝑣⟩)
2		fveq2 6906	. . . . . 6 ⊢ (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘⟨𝑤, 𝑣⟩))
3		df-ov 7433	. . . . . . 7 ⊢ (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}𝑣) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘⟨𝑤, 𝑣⟩)
4		simpl 482	. . . . . . . . 9 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → 𝑥 = 𝑤)
5		mpov 7544	. . . . . . . . . 10 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝑥) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}
6	5	eqcomi 2743	. . . . . . . . 9 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝑥)
7		vex 3481	. . . . . . . . 9 ⊢ 𝑤 ∈ V
8	4, 6, 7	ovmpoa 7587	. . . . . . . 8 ⊢ ((𝑤 ∈ V ∧ 𝑣 ∈ V) → (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}𝑣) = 𝑤)
9	8	el2v 3484	. . . . . . 7 ⊢ (𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}𝑣) = 𝑤
10	3, 9	eqtr3i 2764	. . . . . 6 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘⟨𝑤, 𝑣⟩) = 𝑤
11	2, 10	eqtrdi 2790	. . . . 5 ⊢ (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = 𝑤)
12		vex 3481	. . . . . 6 ⊢ 𝑣 ∈ V
13	7, 12	op1std 8022	. . . . 5 ⊢ (𝐴 = ⟨𝑤, 𝑣⟩ → (1st ‘𝐴) = 𝑤)
14	11, 13	eqtr4d 2777	. . . 4 ⊢ (𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = (1st ‘𝐴))
15	14	exlimivv 1929	. . 3 ⊢ (∃𝑤∃𝑣 𝐴 = ⟨𝑤, 𝑣⟩ → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = (1st ‘𝐴))
16	1, 15	sylbi 217	. 2 ⊢ (𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = (1st ‘𝐴))
17		vex 3481	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
18		vex 3481	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
19	17, 18	pm3.2i 470	. . . . . . . . 9 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
20		ax6ev 1966	. . . . . . . . 9 ⊢ ∃𝑧 𝑧 = 𝑥
21	19, 20	2th 264	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ↔ ∃𝑧 𝑧 = 𝑥)
22	21	opabbii 5214	. . . . . . 7 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 𝑧 = 𝑥}
23		df-xp 5694	. . . . . . 7 ⊢ (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
24		dmoprab 7534	. . . . . . 7 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 𝑧 = 𝑥}
25	22, 23, 24	3eqtr4ri 2773	. . . . . 6 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (V × V)
26	25	eleq2i 2830	. . . . 5 ⊢ (𝐴 ∈ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} ↔ 𝐴 ∈ (V × V))
27		ndmfv 6941	. . . . 5 ⊢ (¬ 𝐴 ∈ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = ∅)
28	26, 27	sylnbir 331	. . . 4 ⊢ (¬ 𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = ∅)
29		dmsnn0 6228	. . . . . . . 8 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
30	29	biimpri 228	. . . . . . 7 ⊢ (dom {𝐴} ≠ ∅ → 𝐴 ∈ (V × V))
31	30	necon1bi 2966	. . . . . 6 ⊢ (¬ 𝐴 ∈ (V × V) → dom {𝐴} = ∅)
32	31	unieqd 4924	. . . . 5 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ dom {𝐴} = ∪ ∅)
33		uni0 4939	. . . . 5 ⊢ ∪ ∅ = ∅
34	32, 33	eqtrdi 2790	. . . 4 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ dom {𝐴} = ∅)
35	28, 34	eqtr4d 2777	. . 3 ⊢ (¬ 𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = ∪ dom {𝐴})
36		1stval 8014	. . 3 ⊢ (1st ‘𝐴) = ∪ dom {𝐴}
37	35, 36	eqtr4di 2792	. 2 ⊢ (¬ 𝐴 ∈ (V × V) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = (1st ‘𝐴))
38	16, 37	pm2.61i 182	1 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = (1st ‘𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1536  ∃wex 1775   ∈ wcel 2105   ≠ wne 2937  Vcvv 3477  ∅c0 4338  {csn 4630  ⟨cop 4636  ∪ cuni 4911  {copab 5209   × cxp 5686  dom cdm 5688  ‘cfv 6562  (class class class)co 7430  {coprab 7431   ∈ cmpo 7432  1^st c1st 8010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012

This theorem is referenced by: (None)