Theorem elmpocl 6047

Description: If a two-parameter class is inhabited, 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 5858	. . . . . 6 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
3	1, 2	eqtri 2191	. . . . 5 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
4	3	dmeqi 4812	. . . 4 ⊢ dom 𝐹 = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
5		dmoprabss 5935	. . . 4 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} ⊆ (𝐴 × 𝐵)
6	4, 5	eqsstri 3179	. . 3 ⊢ dom 𝐹 ⊆ (𝐴 × 𝐵)
7	1	mpofun 5955	. . . . . 6 ⊢ Fun 𝐹
8		funrel 5215	. . . . . 6 ⊢ (Fun 𝐹 → Rel 𝐹)
9	7, 8	ax-mp 5	. . . . 5 ⊢ Rel 𝐹
10		relelfvdm 5528	. . . . 5 ⊢ ((Rel 𝐹 ∧ 𝑋 ∈ (𝐹‘⟨𝑆, 𝑇⟩)) → ⟨𝑆, 𝑇⟩ ∈ dom 𝐹)
11	9, 10	mpan 422	. . . 4 ⊢ (𝑋 ∈ (𝐹‘⟨𝑆, 𝑇⟩) → ⟨𝑆, 𝑇⟩ ∈ dom 𝐹)
12		df-ov 5856	. . . 4 ⊢ (𝑆𝐹𝑇) = (𝐹‘⟨𝑆, 𝑇⟩)
13	11, 12	eleq2s 2265	. . 3 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → ⟨𝑆, 𝑇⟩ ∈ dom 𝐹)
14	6, 13	sselid 3145	. 2 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → ⟨𝑆, 𝑇⟩ ∈ (𝐴 × 𝐵))
15		opelxp 4641	. 2 ⊢ (⟨𝑆, 𝑇⟩ ∈ (𝐴 × 𝐵) ↔ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐵))
16	14, 15	sylib 121	1 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348   ∈ wcel 2141  ⟨cop 3586   × cxp 4609  dom cdm 4611  Rel wrel 4616  Fun wfun 5192  ‘cfv 5198  (class class class)co 5853  {coprab 5854   ∈ cmpo 5855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858

This theorem is referenced by:  elmpocl1  6048  elmpocl2  6049  elovmpo  6050  elpmi  6645  elmapex  6647  pmsspw  6661  ixxssxr  9857  elixx3g  9858  ixxssixx  9859  eliooxr  9884  elfz2  9972  restsspw  12589  ismhm  12685  restrcl  12961  ssrest  12976  iscn2  12994  ishmeo  13098  limcrcl  13421