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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 B𝐶)
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.
StepHypRef Expression
1 elmpt2cl.f . . . . . 6 𝐹 = (x A, y B𝐶)
2 df-mpt2 5441 . . . . . 6 (x A, y B𝐶) = {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) z = 𝐶)}
31, 2eqtri 2042 . . . . 5 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) z = 𝐶)}
43dmeqi 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)
64, 5eqsstri 2952 . . 3 dom 𝐹 ⊆ (A × B)
71mpt2fun 5526 . . . . . 6 Fun 𝐹
8 funrel 4845 . . . . . 6 (Fun 𝐹 → Rel 𝐹)
97, 8ax-mp 7 . . . . 5 Rel 𝐹
10 relelfvdm 5130 . . . . 5 ((Rel 𝐹 𝑋 (𝐹‘⟨𝑆, 𝑇⟩)) → ⟨𝑆, 𝑇 dom 𝐹)
119, 10mpan 402 . . . 4 (𝑋 (𝐹‘⟨𝑆, 𝑇⟩) → ⟨𝑆, 𝑇 dom 𝐹)
12 df-ov 5439 . . . 4 (𝑆𝐹𝑇) = (𝐹‘⟨𝑆, 𝑇⟩)
1311, 12eleq2s 2114 . . 3 (𝑋 (𝑆𝐹𝑇) → ⟨𝑆, 𝑇 dom 𝐹)
146, 13sseldi 2920 . 2 (𝑋 (𝑆𝐹𝑇) → ⟨𝑆, 𝑇 (A × B))
15 opelxp 4301 . 2 (⟨𝑆, 𝑇 (A × B) ↔ (𝑆 A 𝑇 B))
1614, 15sylib 127 1 (𝑋 (𝑆𝐹𝑇) → (𝑆 A 𝑇 B))
Colors of variables: wff set class
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
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