Theorem elrnmpt2res 6672

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 2614	. . . . . 6 ⊢ (𝑧 = 𝐷 → (𝑧 = 𝐶 ↔ 𝐷 = 𝐶))
2	1	anbi1d 737	. . . . 5 ⊢ (𝑧 = 𝐷 → ((𝑧 = 𝐶 ∧ 𝑥𝑅𝑦) ↔ (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦)))
3	2	anbi2d 736	. . . 4 ⊢ (𝑧 = 𝐷 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦))))
4	3	2exbidv 1839	. . 3 ⊢ (𝑧 = 𝐷 → (∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦)) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦))))
5		an12 834	. . . . . . . . . 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 834	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ 𝑅 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑝 ∈ 𝑅 ∧ 𝑧 = 𝐶)))
7		ancom 465	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 𝐶 ∧ 𝑝 ∈ 𝑅) ↔ (𝑝 ∈ 𝑅 ∧ 𝑧 = 𝐶))
8		eleq1 2676	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝑝 ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
9		df-br 4584	. . . . . . . . . . . . . . . 16 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
10	8, 9	syl6bbr 277	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝑝 ∈ 𝑅 ↔ 𝑥𝑅𝑦))
11	10	anbi2d 736	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑧 = 𝐶 ∧ 𝑝 ∈ 𝑅) ↔ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦)))
12	7, 11	syl5bbr 273	. . . . . . . . . . . . 13 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑝 ∈ 𝑅 ∧ 𝑧 = 𝐶) ↔ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦)))
13	12	anbi2d 736	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑝 ∈ 𝑅 ∧ 𝑧 = 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))))
14	6, 13	syl5bb 271	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑝 ∈ 𝑅 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))))
15	14	pm5.32i 667	. . . . . . . . . 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 263	. . . . . . . . 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 1765	. . . . . . . 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 1907	. . . . . . . 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 265	. . . . . . 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 4649	. . . . . 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 6599	. . . . . 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 6554	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
24		dfoprab2 6599	. . . . . . . . 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 2636	. . . . . . . 8 ⊢ 𝐹 = {⟨𝑝, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))}
26	25	reseq1i 5313	. . . . . . 7 ⊢ (𝐹 ↾ 𝑅) = ({⟨𝑝, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))} ↾ 𝑅)
27		resopab 5366	. . . . . . 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 2632	. . . . . 6 ⊢ (𝐹 ↾ 𝑅) = {⟨𝑝, 𝑧⟩ ∣ (𝑝 ∈ 𝑅 ∧ ∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)))}
29	20, 21, 28	3eqtr4ri 2643	. . . . 5 ⊢ (𝐹 ↾ 𝑅) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))}
30	29	rneqi 5273	. . . 4 ⊢ ran (𝐹 ↾ 𝑅) = ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))}
31		rnoprab 6641	. . . 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 2632	. . 3 ⊢ ran (𝐹 ↾ 𝑅) = {𝑧 ∣ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))}
33	4, 32	elab2g 3322	. 2 ⊢ (𝐷 ∈ 𝑉 → (𝐷 ∈ ran (𝐹 ↾ 𝑅) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦))))
34		r2ex 3043	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦)))
35	33, 34	syl6bbr 277	1 ⊢ (𝐷 ∈ 𝑉 → (𝐷 ∈ ran (𝐹 ↾ 𝑅) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∃wrex 2897  ⟨cop 4131   class class class wbr 4583  {copab 4642  ran crn 5039   ↾ cres 5040  {coprab 6550   ↦ cmpt2 6551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-oprab 6553  df-mpt2 6554

This theorem is referenced by: (None)