Theorem elmpt2cl 5621

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

Hypothesis

Ref	Expression
elmpt2cl.f	⊢ 𝐹 = (x ∈ A, y ∈ B ↦ 𝐶)

Assertion

Ref	Expression
elmpt2cl	⊢ (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 elmpt2cl

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elmpt2cl.f	. . . . . 6 ⊢ 𝐹 = (x ∈ A, y ∈ B ↦ 𝐶)
2		df-mpt2 5441	. . . . . 6 ⊢ (x ∈ A, y ∈ B ↦ 𝐶) = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ z = 𝐶)}
3	1, 2	eqtri 2042	. . . . 5 ⊢ 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ z = 𝐶)}
4	3	dmeqi 4463	. . . 4 ⊢ dom 𝐹 = dom {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ z = 𝐶)}
5		dmoprabss 5509	. . . 4 ⊢ dom {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ z = 𝐶)} ⊆ (A × B)
6	4, 5	eqsstri 2952	. . 3 ⊢ dom 𝐹 ⊆ (A × B)
7	1	mpt2fun 5526	. . . . . 6 ⊢ Fun 𝐹
8		funrel 4845	. . . . . 6 ⊢ (Fun 𝐹 → Rel 𝐹)
9	7, 8	ax-mp 7	. . . . 5 ⊢ Rel 𝐹
10		relelfvdm 5130	. . . . 5 ⊢ ((Rel 𝐹 ∧ 𝑋 ∈ (𝐹‘⟨𝑆, 𝑇⟩)) → ⟨𝑆, 𝑇⟩ ∈ dom 𝐹)
11	9, 10	mpan 402	. . . 4 ⊢ (𝑋 ∈ (𝐹‘⟨𝑆, 𝑇⟩) → ⟨𝑆, 𝑇⟩ ∈ dom 𝐹)
12		df-ov 5439	. . . 4 ⊢ (𝑆𝐹𝑇) = (𝐹‘⟨𝑆, 𝑇⟩)
13	11, 12	eleq2s 2114	. . 3 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → ⟨𝑆, 𝑇⟩ ∈ dom 𝐹)
14	6, 13	sseldi 2920	. 2 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → ⟨𝑆, 𝑇⟩ ∈ (A × B))
15		opelxp 4301	. 2 ⊢ (⟨𝑆, 𝑇⟩ ∈ (A × B) ↔ (𝑆 ∈ A ∧ 𝑇 ∈ B))
16	14, 15	sylib 127	1 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → (𝑆 ∈ A ∧ 𝑇 ∈ B))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1228   ∈ wcel 1374  ⟨cop 3353   × cxp 4270  dom cdm 4272  Rel wrel 4277  Fun wfun 4823  ‘cfv 4829  (class class class)co 5436  {coprab 5437   ↦ cmpt2 5438

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-iota 4794  df-fun 4831  df-fv 4837  df-ov 5439  df-oprab 5440  df-mpt2 5441

This theorem is referenced by:  elmpt2cl1  5622  elmpt2cl2  5623  elovmpt2  5624