Theorem dfrn5 36247

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 2197	. . . 4 ⊢ (∃𝑦∃𝑝∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ ∃𝑝∃𝑦∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
2		opex 5447	. . . . . . . 8 ⊢ ⟨𝑦, 𝑧⟩ ∈ V
3		breq1 5113	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑦, 𝑧⟩ → (𝑝2nd 𝑥 ↔ ⟨𝑦, 𝑧⟩2nd 𝑥))
4		eleq1 2851	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑦, 𝑧⟩ → (𝑝 ∈ 𝐴 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
5	3, 4	anbi12d 643	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑦, 𝑧⟩ → ((𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴) ↔ (⟨𝑦, 𝑧⟩2nd 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴)))
6		vex 3459	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
7		vex 3459	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
8	6, 7	br2ndeq 36245	. . . . . . . . . . 11 ⊢ (⟨𝑦, 𝑧⟩2nd 𝑥 ↔ 𝑥 = 𝑧)
9		equcom 2048	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
10	8, 9	bitri 278	. . . . . . . . . 10 ⊢ (⟨𝑦, 𝑧⟩2nd 𝑥 ↔ 𝑧 = 𝑥)
11	10	anbi1i 635	. . . . . . . . 9 ⊢ ((⟨𝑦, 𝑧⟩2nd 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴) ↔ (𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
12	5, 11	bitrdi 290	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑦, 𝑧⟩ → ((𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴) ↔ (𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴)))
13	2, 12	ceqsexv 3503	. . . . . . 7 ⊢ (∃𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ (𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
14	13	exbii 1878	. . . . . 6 ⊢ (∃𝑧∃𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ ∃𝑧(𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
15		excom 2197	. . . . . 6 ⊢ (∃𝑧∃𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ ∃𝑝∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
16		vex 3459	. . . . . . 7 ⊢ 𝑥 ∈ V
17		opeq2 4840	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑥⟩)
18	17	eleq1d 2848	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (⟨𝑦, 𝑧⟩ ∈ 𝐴 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴))
19	16, 18	ceqsexv 3503	. . . . . 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 1878	. . . 4 ⊢ (∃𝑦⟨𝑦, 𝑥⟩ ∈ 𝐴 ↔ ∃𝑦∃𝑝∃𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
22		ancom 465	. . . . . 6 ⊢ ((𝑝 ∈ 𝐴 ∧ 𝑝(2nd ↾ (V × V))𝑥) ↔ (𝑝(2nd ↾ (V × V))𝑥 ∧ 𝑝 ∈ 𝐴))
23		anass 473	. . . . . . 7 ⊢ (((∃𝑦∃𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ 𝑝2nd 𝑥) ∧ 𝑝 ∈ 𝐴) ↔ (∃𝑦∃𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥 ∧ 𝑝 ∈ 𝐴)))
24	16	brresi 5989	. . . . . . . . 9 ⊢ (𝑝(2nd ↾ (V × V))𝑥 ↔ (𝑝 ∈ (V × V) ∧ 𝑝2nd 𝑥))
25		elvv 5738	. . . . . . . . . 10 ⊢ (𝑝 ∈ (V × V) ↔ ∃𝑦∃𝑧 𝑝 = ⟨𝑦, 𝑧⟩)
26	25	anbi1i 635	. . . . . . . . 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 635	. . . . . . 7 ⊢ ((𝑝(2nd ↾ (V × V))𝑥 ∧ 𝑝 ∈ 𝐴) ↔ ((∃𝑦∃𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ 𝑝2nd 𝑥) ∧ 𝑝 ∈ 𝐴))
29		19.41vv 1980	. . . . . . 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 1878	. . . 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 5884	. . 3 ⊢ (𝑥 ∈ ran 𝐴 ↔ ∃𝑦⟨𝑦, 𝑥⟩ ∈ 𝐴)
35	16	elima2 6070	. . 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 2760	1 ⊢ ran 𝐴 = ((2nd ↾ (V × V)) “ 𝐴)

Syntax hints:   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  Vcvv 3455  ⟨cop 4596   class class class wbr 5110   × cxp 5661  ran crn 5664   ↾ cres 5665   “ cima 5666  2^nd c2nd 7986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-fv 6546  df-2nd 7988

This theorem is referenced by:  brrange  36405