Theorem dfrn5 33130

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 2166	. . . 4 ⊢ (∃𝑦∃𝑝∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ ∃𝑝∃𝑦∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
2		opex 5321	. . . . . . . 8 ⊢ ⟨𝑦, 𝑧⟩ ∈ V
3		breq1 5033	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑦, 𝑧⟩ → (𝑝2nd 𝑥 ↔ ⟨𝑦, 𝑧⟩2nd 𝑥))
4		eleq1 2877	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑦, 𝑧⟩ → (𝑝 ∈ 𝐴 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
5	3, 4	anbi12d 633	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑦, 𝑧⟩ → ((𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴) ↔ (⟨𝑦, 𝑧⟩2nd 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴)))
6		vex 3444	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
7		vex 3444	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
8	6, 7	br2ndeq 33128	. . . . . . . . . . 11 ⊢ (⟨𝑦, 𝑧⟩2nd 𝑥 ↔ 𝑥 = 𝑧)
9		equcom 2025	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
10	8, 9	bitri 278	. . . . . . . . . 10 ⊢ (⟨𝑦, 𝑧⟩2nd 𝑥 ↔ 𝑧 = 𝑥)
11	10	anbi1i 626	. . . . . . . . 9 ⊢ ((⟨𝑦, 𝑧⟩2nd 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴) ↔ (𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
12	5, 11	syl6bb 290	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑦, 𝑧⟩ → ((𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴) ↔ (𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴)))
13	2, 12	ceqsexv 3489	. . . . . . 7 ⊢ (∃𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ (𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
14	13	exbii 1849	. . . . . 6 ⊢ (∃𝑧∃𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ ∃𝑧(𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
15		excom 2166	. . . . . 6 ⊢ (∃𝑧∃𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ ∃𝑝∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
16		vex 3444	. . . . . . 7 ⊢ 𝑥 ∈ V
17		opeq2 4765	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑥⟩)
18	17	eleq1d 2874	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (⟨𝑦, 𝑧⟩ ∈ 𝐴 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴))
19	16, 18	ceqsexv 3489	. . . . . 6 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴) ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
20	14, 15, 19	3bitr3ri 305	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ↔ ∃𝑝∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
21	20	exbii 1849	. . . 4 ⊢ (∃𝑦⟨𝑦, 𝑥⟩ ∈ 𝐴 ↔ ∃𝑦∃𝑝∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
22		ancom 464	. . . . . 6 ⊢ ((𝑝 ∈ 𝐴 ∧ 𝑝(2nd ↾ (V × V))𝑥) ↔ (𝑝(2nd ↾ (V × V))𝑥 ∧ 𝑝 ∈ 𝐴))
23		anass 472	. . . . . . 7 ⊢ (((∃𝑦∃𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ 𝑝2nd 𝑥) ∧ 𝑝 ∈ 𝐴) ↔ (∃𝑦∃𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
24	16	brresi 5827	. . . . . . . . 9 ⊢ (𝑝(2nd ↾ (V × V))𝑥 ↔ (𝑝 ∈ (V × V) ∧ 𝑝2nd 𝑥))
25		elvv 5590	. . . . . . . . . 10 ⊢ (𝑝 ∈ (V × V) ↔ ∃𝑦∃𝑧 𝑝 = ⟨𝑦, 𝑧⟩)
26	25	anbi1i 626	. . . . . . . . 9 ⊢ ((𝑝 ∈ (V × V) ∧ 𝑝2nd 𝑥) ↔ (∃𝑦∃𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ 𝑝2nd 𝑥))
27	24, 26	bitri 278	. . . . . . . 8 ⊢ (𝑝(2nd ↾ (V × V))𝑥 ↔ (∃𝑦∃𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ 𝑝2nd 𝑥))
28	27	anbi1i 626	. . . . . . 7 ⊢ ((𝑝(2nd ↾ (V × V))𝑥 ∧ 𝑝 ∈ 𝐴) ↔ ((∃𝑦∃𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ 𝑝2nd 𝑥) ∧ 𝑝 ∈ 𝐴))
29		19.41vv 1951	. . . . . . 7 ⊢ (∃𝑦∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ (∃𝑦∃𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
30	23, 28, 29	3bitr4i 306	. . . . . 6 ⊢ ((𝑝(2nd ↾ (V × V))𝑥 ∧ 𝑝 ∈ 𝐴) ↔ ∃𝑦∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
31	22, 30	bitri 278	. . . . 5 ⊢ ((𝑝 ∈ 𝐴 ∧ 𝑝(2nd ↾ (V × V))𝑥) ↔ ∃𝑦∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
32	31	exbii 1849	. . . 4 ⊢ (∃𝑝(𝑝 ∈ 𝐴 ∧ 𝑝(2nd ↾ (V × V))𝑥) ↔ ∃𝑝∃𝑦∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
33	1, 21, 32	3bitr4i 306	. . 3 ⊢ (∃𝑦⟨𝑦, 𝑥⟩ ∈ 𝐴 ↔ ∃𝑝(𝑝 ∈ 𝐴 ∧ 𝑝(2nd ↾ (V × V))𝑥))
34	16	elrn2 5785	. . 3 ⊢ (𝑥 ∈ ran 𝐴 ↔ ∃𝑦⟨𝑦, 𝑥⟩ ∈ 𝐴)
35	16	elima2 5902	. . 3 ⊢ (𝑥 ∈ ((2nd ↾ (V × V)) “ 𝐴) ↔ ∃𝑝(𝑝 ∈ 𝐴 ∧ 𝑝(2nd ↾ (V × V))𝑥))
36	33, 34, 35	3bitr4i 306	. 2 ⊢ (𝑥 ∈ ran 𝐴 ↔ 𝑥 ∈ ((2nd ↾ (V × V)) “ 𝐴))
37	36	eqriv 2795	1 ⊢ ran 𝐴 = ((2nd ↾ (V × V)) “ 𝐴)

Syntax hints:   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3441  ⟨cop 4531   class class class wbr 5030   × cxp 5517  ran crn 5520   ↾ cres 5521   “ cima 5522  2^nd c2nd 7670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fo 6330  df-fv 6332  df-2nd 7672

This theorem is referenced by:  brrange  33508