Theorem elrnmpores 7493

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 2740	. . . . . 6 ⊢ (𝑧 = 𝐷 → (𝑧 = 𝐶 ↔ 𝐷 = 𝐶))
2	1	anbi1d 630	. . . . 5 ⊢ (𝑧 = 𝐷 → ((𝑧 = 𝐶 ∧ 𝑥𝑅𝑦) ↔ (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦)))
3	2	anbi2d 629	. . . 4 ⊢ (𝑧 = 𝐷 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦))))
4	3	2exbidv 1927	. . 3 ⊢ (𝑧 = 𝐷 → (∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦)) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦))))
5		an12 643	. . . . . . . . . 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 643	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ 𝑅 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑝 ∈ 𝑅 ∧ 𝑧 = 𝐶)))
7		ancom 461	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 𝐶 ∧ 𝑝 ∈ 𝑅) ↔ (𝑝 ∈ 𝑅 ∧ 𝑧 = 𝐶))
8		eleq1 2825	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝑝 ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
9		df-br 5106	. . . . . . . . . . . . . . . 16 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
10	8, 9	bitr4di 288	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝑝 ∈ 𝑅 ↔ 𝑥𝑅𝑦))
11	10	anbi2d 629	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑧 = 𝐶 ∧ 𝑝 ∈ 𝑅) ↔ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦)))
12	7, 11	bitr3id 284	. . . . . . . . . . . . 13 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑝 ∈ 𝑅 ∧ 𝑧 = 𝐶) ↔ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦)))
13	12	anbi2d 629	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑝 ∈ 𝑅 ∧ 𝑧 = 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))))
14	6, 13	bitrid 282	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑝 ∈ 𝑅 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))))
15	14	pm5.32i 575	. . . . . . . . . 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 274	. . . . . . . . 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 1851	. . . . . . . 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 1961	. . . . . . . 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 276	. . . . . . 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 5172	. . . . . 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 7415	. . . . . 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 7362	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
24		dfoprab2 7415	. . . . . . . . 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 5988	. . . . . . 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 5892	. . . 4 ⊢ ran (𝐹 ↾ 𝑅) = ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑥𝑅𝑦))}
31		rnoprab 7460	. . . 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 3632	. 2 ⊢ (𝐷 ∈ 𝑉 → (𝐷 ∈ ran (𝐹 ↾ 𝑅) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦))))
34		r2ex 3192	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦)))
35	33, 34	bitr4di 288	1 ⊢ (𝐷 ∈ 𝑉 → (𝐷 ∈ ran (𝐹 ↾ 𝑅) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  {cab 2713  ∃wrex 3073  ⟨cop 4592   class class class wbr 5105  {copab 5167  ran crn 5634   ↾ cres 5635  {coprab 7358   ∈ cmpo 7359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-xp 5639  df-rel 5640  df-cnv 5641  df-dm 5643  df-rn 5644  df-res 5645  df-oprab 7361  df-mpo 7362

This theorem is referenced by: (None)