Theorem elrnmpores 7497

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
elrnmpores	⊢ (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 elrnmpores

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

Step	Hyp	Ref	Expression
1		eqeq1 2745	. . . . . 6 ⊢ (𝑧 = 𝐷 → (𝑧 = 𝐶 ↔ 𝐷 = 𝐶))
2	1	anbi1d 638	. . . . 5 ⊢ (𝑧 = 𝐷 → ((𝑧 = 𝐶 ∧ 𝑥𝑅𝑦) ↔ (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦)))
3	2	anbi2d 637	. . . 4 ⊢ (𝑧 = 𝐷 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦))))
4	3	2exbidv 1932	. . 3 ⊢ (𝑧 = 𝐷 → (∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦)) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦))))
5		an12 652	. . . . . . . . . 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 652	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ 𝑅 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑝 ∈ 𝑅 ∧ 𝑧 = 𝐶)))
7		ancom 462	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 𝐶 ∧ 𝑝 ∈ 𝑅) ↔ (𝑝 ∈ 𝑅 ∧ 𝑧 = 𝐶))
8		eleq1 2829	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝑝 ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
9		df-br 5075	. . . . . . . . . . . . . . . 16 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
10	8, 9	bitr4di 291	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝑝 ∈ 𝑅 ↔ 𝑥𝑅𝑦))
11	10	anbi2d 637	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑧 = 𝐶 ∧ 𝑝 ∈ 𝑅) ↔ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦)))
12	7, 11	bitr3id 287	. . . . . . . . . . . . 13 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑝 ∈ 𝑅 ∧ 𝑧 = 𝐶) ↔ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦)))
13	12	anbi2d 637	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑝 ∈ 𝑅 ∧ 𝑧 = 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))))
14	6, 13	bitrid 285	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑝 ∈ 𝑅 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))))
15	14	pm5.32i 580	. . . . . . . . . 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 277	. . . . . . . . 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 1857	. . . . . . . 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 1965	. . . . . . . 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 279	. . . . . . 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 5141	. . . . . 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 7417	. . . . . 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-mpo 7364	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
24		dfoprab2 7417	. . . . . . . . 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 2768	. . . . . . . 8 ⊢ 𝐹 = {⟨𝑝, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))}
26	25	reseq1i 5933	. . . . . . 7 ⊢ (𝐹 ↾ 𝑅) = ({⟨𝑝, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))} ↾ 𝑅)
27		resopab 5992	. . . . . . 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 2764	. . . . . 6 ⊢ (𝐹 ↾ 𝑅) = {⟨𝑝, 𝑧⟩ ∣ (𝑝 ∈ 𝑅 ∧ ∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)))}
29	20, 21, 28	3eqtr4ri 2775	. . . . 5 ⊢ (𝐹 ↾ 𝑅) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))}
30	29	rneqi 5885	. . . 4 ⊢ ran (𝐹 ↾ 𝑅) = ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))}
31		rnoprab 7464	. . . 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 2764	. . 3 ⊢ ran (𝐹 ↾ 𝑅) = {𝑧 ∣ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))}
33	4, 32	elab2g 3619	. 2 ⊢ (𝐷 ∈ 𝑉 → (𝐷 ∈ ran (𝐹 ↾ 𝑅) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦))))
34		r2ex 3178	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦)))
35	33, 34	bitr4di 291	1 ⊢ (𝐷 ∈ 𝑉 → (𝐷 ∈ ran (𝐹 ↾ 𝑅) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548  ∃wex 1787   ∈ wcel 2121  {cab 2719  ∃wrex 3065  ⟨cop 4563   class class class wbr 5074  {copab 5136  ran crn 5621   ↾ cres 5622  {coprab 7360   ∈ cmpo 7361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5220  ax-pr 5364

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-br 5075  df-opab 5137  df-xp 5626  df-rel 5627  df-cnv 5628  df-dm 5630  df-rn 5631  df-res 5632  df-oprab 7363  df-mpo 7364

This theorem is referenced by: (None)