Theorem dfrn5 33012

Description: Definition of range in terms of 2^nd and image. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)

Assertion

Ref	Expression
dfrn5	⊢ ran 𝐴 = ((2nd ↾ (V × V)) “ 𝐴)

Proof of Theorem dfrn5

Dummy variables 𝑝 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		excom 2165	. . . 4 ⊢ (∃𝑦∃𝑝∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ ∃𝑝∃𝑦∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
2		opex 5348	. . . . . . . 8 ⊢ ⟨𝑦, 𝑧⟩ ∈ V
3		breq1 5061	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑦, 𝑧⟩ → (𝑝2nd 𝑥 ↔ ⟨𝑦, 𝑧⟩2nd 𝑥))
4		eleq1 2900	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑦, 𝑧⟩ → (𝑝 ∈ 𝐴 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
5	3, 4	anbi12d 632	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑦, 𝑧⟩ → ((𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴) ↔ (⟨𝑦, 𝑧⟩2nd 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴)))
6		vex 3497	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
7		vex 3497	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
8	6, 7	br2ndeq 33010	. . . . . . . . . . 11 ⊢ (⟨𝑦, 𝑧⟩2nd 𝑥 ↔ 𝑥 = 𝑧)
9		equcom 2021	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
10	8, 9	bitri 277	. . . . . . . . . 10 ⊢ (⟨𝑦, 𝑧⟩2nd 𝑥 ↔ 𝑧 = 𝑥)
11	10	anbi1i 625	. . . . . . . . 9 ⊢ ((⟨𝑦, 𝑧⟩2nd 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴) ↔ (𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
12	5, 11	syl6bb 289	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑦, 𝑧⟩ → ((𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴) ↔ (𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴)))
13	2, 12	ceqsexv 3541	. . . . . . 7 ⊢ (∃𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ (𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
14	13	exbii 1844	. . . . . 6 ⊢ (∃𝑧∃𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ ∃𝑧(𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
15		excom 2165	. . . . . 6 ⊢ (∃𝑧∃𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ ∃𝑝∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
16		vex 3497	. . . . . . 7 ⊢ 𝑥 ∈ V
17		opeq2 4797	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑥⟩)
18	17	eleq1d 2897	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (⟨𝑦, 𝑧⟩ ∈ 𝐴 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴))
19	16, 18	ceqsexv 3541	. . . . . 6 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴) ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
20	14, 15, 19	3bitr3ri 304	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ↔ ∃𝑝∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
21	20	exbii 1844	. . . 4 ⊢ (∃𝑦⟨𝑦, 𝑥⟩ ∈ 𝐴 ↔ ∃𝑦∃𝑝∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
22		ancom 463	. . . . . 6 ⊢ ((𝑝 ∈ 𝐴 ∧ 𝑝(2nd ↾ (V × V))𝑥) ↔ (𝑝(2nd ↾ (V × V))𝑥 ∧ 𝑝 ∈ 𝐴))
23		anass 471	. . . . . . 7 ⊢ (((∃𝑦∃𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ 𝑝2nd 𝑥) ∧ 𝑝 ∈ 𝐴) ↔ (∃𝑦∃𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
24	16	brresi 5856	. . . . . . . . 9 ⊢ (𝑝(2nd ↾ (V × V))𝑥 ↔ (𝑝 ∈ (V × V) ∧ 𝑝2nd 𝑥))
25		elvv 5620	. . . . . . . . . 10 ⊢ (𝑝 ∈ (V × V) ↔ ∃𝑦∃𝑧 𝑝 = ⟨𝑦, 𝑧⟩)
26	25	anbi1i 625	. . . . . . . . 9 ⊢ ((𝑝 ∈ (V × V) ∧ 𝑝2nd 𝑥) ↔ (∃𝑦∃𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ 𝑝2nd 𝑥))
27	24, 26	bitri 277	. . . . . . . 8 ⊢ (𝑝(2nd ↾ (V × V))𝑥 ↔ (∃𝑦∃𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ 𝑝2nd 𝑥))
28	27	anbi1i 625	. . . . . . 7 ⊢ ((𝑝(2nd ↾ (V × V))𝑥 ∧ 𝑝 ∈ 𝐴) ↔ ((∃𝑦∃𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ 𝑝2nd 𝑥) ∧ 𝑝 ∈ 𝐴))
29		19.41vv 1947	. . . . . . 7 ⊢ (∃𝑦∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ (∃𝑦∃𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
30	23, 28, 29	3bitr4i 305	. . . . . 6 ⊢ ((𝑝(2nd ↾ (V × V))𝑥 ∧ 𝑝 ∈ 𝐴) ↔ ∃𝑦∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
31	22, 30	bitri 277	. . . . 5 ⊢ ((𝑝 ∈ 𝐴 ∧ 𝑝(2nd ↾ (V × V))𝑥) ↔ ∃𝑦∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
32	31	exbii 1844	. . . 4 ⊢ (∃𝑝(𝑝 ∈ 𝐴 ∧ 𝑝(2nd ↾ (V × V))𝑥) ↔ ∃𝑝∃𝑦∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
33	1, 21, 32	3bitr4i 305	. . 3 ⊢ (∃𝑦⟨𝑦, 𝑥⟩ ∈ 𝐴 ↔ ∃𝑝(𝑝 ∈ 𝐴 ∧ 𝑝(2nd ↾ (V × V))𝑥))
34	16	elrn2 5815	. . 3 ⊢ (𝑥 ∈ ran 𝐴 ↔ ∃𝑦⟨𝑦, 𝑥⟩ ∈ 𝐴)
35	16	elima2 5929	. . 3 ⊢ (𝑥 ∈ ((2nd ↾ (V × V)) “ 𝐴) ↔ ∃𝑝(𝑝 ∈ 𝐴 ∧ 𝑝(2nd ↾ (V × V))𝑥))
36	33, 34, 35	3bitr4i 305	. 2 ⊢ (𝑥 ∈ ran 𝐴 ↔ 𝑥 ∈ ((2nd ↾ (V × V)) “ 𝐴))
37	36	eqriv 2818	1 ⊢ ran 𝐴 = ((2nd ↾ (V × V)) “ 𝐴)

Syntax hints:   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110  Vcvv 3494  ⟨cop 4566   class class class wbr 5058   × cxp 5547  ran crn 5550   ↾ cres 5551   “ cima 5552  2^nd c2nd 7682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fo 6355  df-fv 6357  df-2nd 7684

This theorem is referenced by:  brrange  33390