Theorem reldmmpo 7285

Description: The domain of an operation defined by maps-to notation is a relation. (Contributed by Stefan O'Rear, 27-Nov-2014.)

Hypothesis

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

Assertion

Ref	Expression
reldmmpo	⊢ Rel dom 𝐹

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦

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

Proof of Theorem reldmmpo

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reldmoprab 7259	. 2 ⊢ Rel dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
2		rngop.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
3		df-mpo 7161	. . . . 5 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
4	2, 3	eqtri 2844	. . . 4 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
5	4	dmeqi 5773	. . 3 ⊢ dom 𝐹 = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
6	5	releqi 5652	. 2 ⊢ (Rel dom 𝐹 ↔ Rel dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)})
7	1, 6	mpbir 233	1 ⊢ Rel dom 𝐹

Syntax hints:   ∧ wa 398   = wceq 1537   ∈ wcel 2114  dom cdm 5555  Rel wrel 5560  {coprab 7157   ∈ cmpo 7158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-xp 5561  df-rel 5562  df-dm 5565  df-oprab 7160  df-mpo 7161

This theorem is referenced by:  reldmmap  8415  reldmsets  16511  reldmress  16550  reldmprds  16722  gsum0  17894  reldmghm  18357  oppglsm  18767  reldmdprd  19119  reldmlmhm  19797  reldmpsr  20141  reldmmpl  20207  reldmopsr  20254  reldmevls  20297  vr1val  20360  reldmevls1  20480  evl1fval  20491  zrhval  20655  reldmdsmm  20877  frlmrcl  20901  matbas0pc  21018  mdetfval  21195  madufval  21246  qtopres  22306  fgabs  22487  reldmtng  23247  reldmnghm  23321  reldmnmhm  23322  dvbsss  24500  reldmmdeg  24651  nbgrprc0  27116  wwlksn  27615  reldmresv  30899  bj-restsnid  34381  mzpmfp  39393  brovmptimex  40426