Theorem elrnmpores 7554

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 2738	. . . . . 6 ⊢ (𝑧 = 𝐷 → (𝑧 = 𝐶 ↔ 𝐷 = 𝐶))
2	1	anbi1d 631	. . . . 5 ⊢ (𝑧 = 𝐷 → ((𝑧 = 𝐶 ∧ 𝑥𝑅𝑦) ↔ (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦)))
3	2	anbi2d 630	. . . 4 ⊢ (𝑧 = 𝐷 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦))))
4	3	2exbidv 1923	. . 3 ⊢ (𝑧 = 𝐷 → (∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦)) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦))))
5		an12 645	. . . . . . . . . 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 645	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ 𝑅 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑝 ∈ 𝑅 ∧ 𝑧 = 𝐶)))
7		ancom 460	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 𝐶 ∧ 𝑝 ∈ 𝑅) ↔ (𝑝 ∈ 𝑅 ∧ 𝑧 = 𝐶))
8		eleq1 2821	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝑝 ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
9		df-br 5126	. . . . . . . . . . . . . . . 16 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
10	8, 9	bitr4di 289	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝑝 ∈ 𝑅 ↔ 𝑥𝑅𝑦))
11	10	anbi2d 630	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑧 = 𝐶 ∧ 𝑝 ∈ 𝑅) ↔ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦)))
12	7, 11	bitr3id 285	. . . . . . . . . . . . 13 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑝 ∈ 𝑅 ∧ 𝑧 = 𝐶) ↔ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦)))
13	12	anbi2d 630	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑝 ∈ 𝑅 ∧ 𝑧 = 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))))
14	6, 13	bitrid 283	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑝 ∈ 𝑅 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))))
15	14	pm5.32i 574	. . . . . . . . . 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 275	. . . . . . . . 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 1848	. . . . . . . 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 1956	. . . . . . . 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 277	. . . . . . 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 5192	. . . . . 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 7474	. . . . . 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 7419	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
24		dfoprab2 7474	. . . . . . . . 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 2761	. . . . . . . 8 ⊢ 𝐹 = {⟨𝑝, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))}
26	25	reseq1i 5975	. . . . . . 7 ⊢ (𝐹 ↾ 𝑅) = ({⟨𝑝, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))} ↾ 𝑅)
27		resopab 6034	. . . . . . 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 2757	. . . . . 6 ⊢ (𝐹 ↾ 𝑅) = {⟨𝑝, 𝑧⟩ ∣ (𝑝 ∈ 𝑅 ∧ ∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)))}
29	20, 21, 28	3eqtr4ri 2768	. . . . 5 ⊢ (𝐹 ↾ 𝑅) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))}
30	29	rneqi 5930	. . . 4 ⊢ ran (𝐹 ↾ 𝑅) = ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))}
31		rnoprab 7521	. . . 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 2757	. . 3 ⊢ ran (𝐹 ↾ 𝑅) = {𝑧 ∣ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))}
33	4, 32	elab2g 3664	. 2 ⊢ (𝐷 ∈ 𝑉 → (𝐷 ∈ ran (𝐹 ↾ 𝑅) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦))))
34		r2ex 3183	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦)))
35	33, 34	bitr4di 289	1 ⊢ (𝐷 ∈ 𝑉 → (𝐷 ∈ ran (𝐹 ↾ 𝑅) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2107  {cab 2712  ∃wrex 3059  ⟨cop 4614   class class class wbr 5125  {copab 5187  ran crn 5668   ↾ cres 5669  {coprab 7415   ∈ cmpo 7416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pr 5414

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316  df-if 4508  df-sn 4609  df-pr 4611  df-op 4615  df-br 5126  df-opab 5188  df-xp 5673  df-rel 5674  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679  df-oprab 7418  df-mpo 7419

This theorem is referenced by: (None)