Theorem dfrn5 34745

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 2163	. . . 4 ⊢ (∃𝑦∃𝑝∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ ∃𝑝∃𝑦∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
2		opex 5465	. . . . . . . 8 ⊢ ⟨𝑦, 𝑧⟩ ∈ V
3		breq1 5152	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑦, 𝑧⟩ → (𝑝2nd 𝑥 ↔ ⟨𝑦, 𝑧⟩2nd 𝑥))
4		eleq1 2822	. . . . . . . . . 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 3479	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
7		vex 3479	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
8	6, 7	br2ndeq 34743	. . . . . . . . . . 11 ⊢ (⟨𝑦, 𝑧⟩2nd 𝑥 ↔ 𝑥 = 𝑧)
9		equcom 2022	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
10	8, 9	bitri 275	. . . . . . . . . 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	bitrdi 287	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑦, 𝑧⟩ → ((𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴) ↔ (𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴)))
13	2, 12	ceqsexv 3526	. . . . . . 7 ⊢ (∃𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ (𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
14	13	exbii 1851	. . . . . 6 ⊢ (∃𝑧∃𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ ∃𝑧(𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
15		excom 2163	. . . . . 6 ⊢ (∃𝑧∃𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ ∃𝑝∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
16		vex 3479	. . . . . . 7 ⊢ 𝑥 ∈ V
17		opeq2 4875	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑥⟩)
18	17	eleq1d 2819	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (⟨𝑦, 𝑧⟩ ∈ 𝐴 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴))
19	16, 18	ceqsexv 3526	. . . . . 6 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴) ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
20	14, 15, 19	3bitr3ri 302	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ↔ ∃𝑝∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
21	20	exbii 1851	. . . 4 ⊢ (∃𝑦⟨𝑦, 𝑥⟩ ∈ 𝐴 ↔ ∃𝑦∃𝑝∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
22		ancom 462	. . . . . 6 ⊢ ((𝑝 ∈ 𝐴 ∧ 𝑝(2nd ↾ (V × V))𝑥) ↔ (𝑝(2nd ↾ (V × V))𝑥 ∧ 𝑝 ∈ 𝐴))
23		anass 470	. . . . . . 7 ⊢ (((∃𝑦∃𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ 𝑝2nd 𝑥) ∧ 𝑝 ∈ 𝐴) ↔ (∃𝑦∃𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
24	16	brresi 5991	. . . . . . . . 9 ⊢ (𝑝(2nd ↾ (V × V))𝑥 ↔ (𝑝 ∈ (V × V) ∧ 𝑝2nd 𝑥))
25		elvv 5751	. . . . . . . . . 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 275	. . . . . . . 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 1955	. . . . . . 7 ⊢ (∃𝑦∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ (∃𝑦∃𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
30	23, 28, 29	3bitr4i 303	. . . . . 6 ⊢ ((𝑝(2nd ↾ (V × V))𝑥 ∧ 𝑝 ∈ 𝐴) ↔ ∃𝑦∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
31	22, 30	bitri 275	. . . . 5 ⊢ ((𝑝 ∈ 𝐴 ∧ 𝑝(2nd ↾ (V × V))𝑥) ↔ ∃𝑦∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
32	31	exbii 1851	. . . 4 ⊢ (∃𝑝(𝑝 ∈ 𝐴 ∧ 𝑝(2nd ↾ (V × V))𝑥) ↔ ∃𝑝∃𝑦∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
33	1, 21, 32	3bitr4i 303	. . 3 ⊢ (∃𝑦⟨𝑦, 𝑥⟩ ∈ 𝐴 ↔ ∃𝑝(𝑝 ∈ 𝐴 ∧ 𝑝(2nd ↾ (V × V))𝑥))
34	16	elrn2 5893	. . 3 ⊢ (𝑥 ∈ ran 𝐴 ↔ ∃𝑦⟨𝑦, 𝑥⟩ ∈ 𝐴)
35	16	elima2 6066	. . 3 ⊢ (𝑥 ∈ ((2nd ↾ (V × V)) “ 𝐴) ↔ ∃𝑝(𝑝 ∈ 𝐴 ∧ 𝑝(2nd ↾ (V × V))𝑥))
36	33, 34, 35	3bitr4i 303	. 2 ⊢ (𝑥 ∈ ran 𝐴 ↔ 𝑥 ∈ ((2nd ↾ (V × V)) “ 𝐴))
37	36	eqriv 2730	1 ⊢ ran 𝐴 = ((2nd ↾ (V × V)) “ 𝐴)

Syntax hints:   ∧ wa 397   = wceq 1542  ∃wex 1782   ∈ wcel 2107  Vcvv 3475  ⟨cop 4635   class class class wbr 5149   × cxp 5675  ran crn 5678   ↾ cres 5679   “ cima 5680  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-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-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-2nd 7976

This theorem is referenced by:  brrange  34906