Theorem elmpocl 7543

Description: If a two-parameter class is not empty, constrain the implicit pair. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Hypothesis

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

Assertion

Ref	Expression
elmpocl	⊢ (𝑋 ∈ (𝑆𝐹𝑇) → (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐵))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Allowed substitution hints:   𝐶(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝑇(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑋(𝑥,𝑦)

Proof of Theorem elmpocl

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elmpocl.f	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
2		df-mpo 7312	. . . . . 6 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
3	1, 2	eqtri 2764	. . . . 5 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
4	3	dmeqi 5826	. . . 4 ⊢ dom 𝐹 = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
5		dmoprabss 7409	. . . 4 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} ⊆ (𝐴 × 𝐵)
6	4, 5	eqsstri 3960	. . 3 ⊢ dom 𝐹 ⊆ (𝐴 × 𝐵)
7		elfvdm 6838	. . . 4 ⊢ (𝑋 ∈ (𝐹‘⟨𝑆, 𝑇⟩) → ⟨𝑆, 𝑇⟩ ∈ dom 𝐹)
8		df-ov 7310	. . . 4 ⊢ (𝑆𝐹𝑇) = (𝐹‘⟨𝑆, 𝑇⟩)
9	7, 8	eleq2s 2855	. . 3 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → ⟨𝑆, 𝑇⟩ ∈ dom 𝐹)
10	6, 9	sselid 3924	. 2 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → ⟨𝑆, 𝑇⟩ ∈ (𝐴 × 𝐵))
11		opelxp 5636	. 2 ⊢ (⟨𝑆, 𝑇⟩ ∈ (𝐴 × 𝐵) ↔ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐵))
12	10, 11	sylib 217	1 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1539   ∈ wcel 2104  ⟨cop 4571   × cxp 5598  dom cdm 5600  ‘cfv 6458  (class class class)co 7307  {coprab 7308   ∈ cmpo 7309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3306  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-xp 5606  df-dm 5610  df-iota 6410  df-fv 6466  df-ov 7310  df-oprab 7311  df-mpo 7312

This theorem is referenced by:  elmpocl1  7544  elmpocl2  7545  elovmpo  7546  elovmporab  7547  elovmporab1w  7548  elovmporab1  7549  el2mpocsbcl  7957  ixxssixx  13143  funcrcl  17627  natrcl  17715  ismhm  18481  isghm  18883  isga  18946  isslw  19262  isrhm  20014  rimrcl  20017  islmhm  20338  iscn2  22438  elflim2  23164  isfcls  23209  isnmhm  23959  limcrcl  25087  ewlkprop  28019  wwlknbp  28256  wspthnp  28264  iscvm  33270  mclsrcl  33572  mgmhmrcl  45579  intop  45641  rnghmrcl  45691  rngimrcl  45699  naryrcl  46221