Theorem elrnmpt2res 6774

Description: Membership in the range of a restricted operation class abstraction. (Contributed by Thierry Arnoux, 25-May-2019.)

Hypothesis

Ref	Expression
rngop.1	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Assertion

Ref	Expression
elrnmpt2res	⊢ (𝐷 ∈ 𝑉 → (𝐷 ∈ ran (𝐹 ↾ 𝑅) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦)))

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦,𝐷   𝑥,𝑅,𝑦

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem elrnmpt2res

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

Step	Hyp	Ref	Expression
1		eqeq1 2626	. . . . . 6 ⊢ (𝑧 = 𝐷 → (𝑧 = 𝐶 ↔ 𝐷 = 𝐶))
2	1	anbi1d 741	. . . . 5 ⊢ (𝑧 = 𝐷 → ((𝑧 = 𝐶 ∧ 𝑥𝑅𝑦) ↔ (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦)))
3	2	anbi2d 740	. . . 4 ⊢ (𝑧 = 𝐷 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦))))
4	3	2exbidv 1852	. . 3 ⊢ (𝑧 = 𝐷 → (∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦)) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦))))
5		an12 838	. . . . . . . . . 10 ⊢ ((𝑝 ∈ 𝑅 ∧ (𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))) ↔ (𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑝 ∈ 𝑅 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))))
6		an12 838	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ 𝑅 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑝 ∈ 𝑅 ∧ 𝑧 = 𝐶)))
7		ancom 466	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 𝐶 ∧ 𝑝 ∈ 𝑅) ↔ (𝑝 ∈ 𝑅 ∧ 𝑧 = 𝐶))
8		eleq1 2689	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝑝 ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
9		df-br 4654	. . . . . . . . . . . . . . . 16 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
10	8, 9	syl6bbr 278	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝑝 ∈ 𝑅 ↔ 𝑥𝑅𝑦))
11	10	anbi2d 740	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑧 = 𝐶 ∧ 𝑝 ∈ 𝑅) ↔ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦)))
12	7, 11	syl5bbr 274	. . . . . . . . . . . . 13 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑝 ∈ 𝑅 ∧ 𝑧 = 𝐶) ↔ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦)))
13	12	anbi2d 740	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑝 ∈ 𝑅 ∧ 𝑧 = 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))))
14	6, 13	syl5bb 272	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑝 ∈ 𝑅 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))))
15	14	pm5.32i 669	. . . . . . . . . 10 ⊢ ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑝 ∈ 𝑅 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))) ↔ (𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))))
16	5, 15	bitri 264	. . . . . . . . 9 ⊢ ((𝑝 ∈ 𝑅 ∧ (𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))) ↔ (𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))))
17	16	2exbii 1775	. . . . . . . 8 ⊢ (∃𝑥∃𝑦(𝑝 ∈ 𝑅 ∧ (𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))) ↔ ∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))))
18		19.42vv 1920	. . . . . . . 8 ⊢ (∃𝑥∃𝑦(𝑝 ∈ 𝑅 ∧ (𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))) ↔ (𝑝 ∈ 𝑅 ∧ ∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))))
19	17, 18	bitr3i 266	. . . . . . 7 ⊢ (∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))) ↔ (𝑝 ∈ 𝑅 ∧ ∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))))
20	19	opabbii 4717	. . . . . 6 ⊢ {⟨𝑝, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦)))} = {⟨𝑝, 𝑧⟩ ∣ (𝑝 ∈ 𝑅 ∧ ∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)))}
21		dfoprab2 6701	. . . . . 6 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))} = {⟨𝑝, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦)))}
22		rngop.1	. . . . . . . . 9 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
23		df-mpt2 6655	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
24		dfoprab2 6701	. . . . . . . . 9 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} = {⟨𝑝, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))}
25	22, 23, 24	3eqtri 2648	. . . . . . . 8 ⊢ 𝐹 = {⟨𝑝, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))}
26	25	reseq1i 5392	. . . . . . 7 ⊢ (𝐹 ↾ 𝑅) = ({⟨𝑝, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))} ↾ 𝑅)
27		resopab 5446	. . . . . . 7 ⊢ ({⟨𝑝, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))} ↾ 𝑅) = {⟨𝑝, 𝑧⟩ ∣ (𝑝 ∈ 𝑅 ∧ ∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)))}
28	26, 27	eqtri 2644	. . . . . 6 ⊢ (𝐹 ↾ 𝑅) = {⟨𝑝, 𝑧⟩ ∣ (𝑝 ∈ 𝑅 ∧ ∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)))}
29	20, 21, 28	3eqtr4ri 2655	. . . . 5 ⊢ (𝐹 ↾ 𝑅) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))}
30	29	rneqi 5352	. . . 4 ⊢ ran (𝐹 ↾ 𝑅) = ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))}
31		rnoprab 6743	. . . 4 ⊢ ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))} = {𝑧 ∣ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))}
32	30, 31	eqtri 2644	. . 3 ⊢ ran (𝐹 ↾ 𝑅) = {𝑧 ∣ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))}
33	4, 32	elab2g 3353	. 2 ⊢ (𝐷 ∈ 𝑉 → (𝐷 ∈ ran (𝐹 ↾ 𝑅) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦))))
34		r2ex 3061	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦)))
35	33, 34	syl6bbr 278	1 ⊢ (𝐷 ∈ 𝑉 → (𝐷 ∈ ran (𝐹 ↾ 𝑅) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608  ∃wrex 2913  ⟨cop 4183   class class class wbr 4653  {copab 4712  ran crn 5115   ↾ cres 5116  {coprab 6651   ↦ cmpt2 6652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-oprab 6654  df-mpt2 6655

This theorem is referenced by: (None)