Theorem elmpocl 7387

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 7161	. . . . . 6 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
3	1, 2	eqtri 2844	. . . . 5 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
4	3	dmeqi 5773	. . . 4 ⊢ dom 𝐹 = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
5		dmoprabss 7256	. . . 4 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} ⊆ (𝐴 × 𝐵)
6	4, 5	eqsstri 4001	. . 3 ⊢ dom 𝐹 ⊆ (𝐴 × 𝐵)
7		elfvdm 6702	. . . 4 ⊢ (𝑋 ∈ (𝐹‘⟨𝑆, 𝑇⟩) → ⟨𝑆, 𝑇⟩ ∈ dom 𝐹)
8		df-ov 7159	. . . 4 ⊢ (𝑆𝐹𝑇) = (𝐹‘⟨𝑆, 𝑇⟩)
9	7, 8	eleq2s 2931	. . 3 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → ⟨𝑆, 𝑇⟩ ∈ dom 𝐹)
10	6, 9	sseldi 3965	. 2 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → ⟨𝑆, 𝑇⟩ ∈ (𝐴 × 𝐵))
11		opelxp 5591	. 2 ⊢ (⟨𝑆, 𝑇⟩ ∈ (𝐴 × 𝐵) ↔ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐵))
12	10, 11	sylib 220	1 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ⟨cop 4573   × cxp 5553  dom cdm 5555  ‘cfv 6355  (class class class)co 7156  {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-pow 5266  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-ral 3143  df-rex 3144  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-uni 4839  df-br 5067  df-opab 5129  df-xp 5561  df-dm 5565  df-iota 6314  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161

This theorem is referenced by:  elmpocl1  7388  elmpocl2  7389  elovmpo  7390  elovmporab  7391  elovmporab1w  7392  elovmporab1  7393  el2mpocsbcl  7780  ixxssixx  12753  funcrcl  17133  natrcl  17220  ismhm  17958  isghm  18358  isga  18421  isslw  18733  isrhm  19473  rimrcl  19476  islmhm  19799  iscn2  21846  elflim2  22572  isfcls  22617  isnmhm  23355  limcrcl  24472  ewlkprop  27385  wwlknbp  27620  wspthnp  27628  iscvm  32506  mclsrcl  32808  mgmhmrcl  44097  intop  44159  rnghmrcl  44209  rngimrcl  44217