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Theorem elmpt2cl 5640
 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 5460 . . . . . 6 (x A, y B𝐶) = {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) z = 𝐶)}
31, 2eqtri 2057 . . . . 5 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) z = 𝐶)}
43dmeqi 4479 . . . 4 dom 𝐹 = dom {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) z = 𝐶)}
5 dmoprabss 5528 . . . 4 dom {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) z = 𝐶)} ⊆ (A × B)
64, 5eqsstri 2969 . . 3 dom 𝐹 ⊆ (A × B)
71mpt2fun 5545 . . . . . 6 Fun 𝐹
8 funrel 4862 . . . . . 6 (Fun 𝐹 → Rel 𝐹)
97, 8ax-mp 7 . . . . 5 Rel 𝐹
10 relelfvdm 5148 . . . . 5 ((Rel 𝐹 𝑋 (𝐹‘⟨𝑆, 𝑇⟩)) → ⟨𝑆, 𝑇 dom 𝐹)
119, 10mpan 400 . . . 4 (𝑋 (𝐹‘⟨𝑆, 𝑇⟩) → ⟨𝑆, 𝑇 dom 𝐹)
12 df-ov 5458 . . . 4 (𝑆𝐹𝑇) = (𝐹‘⟨𝑆, 𝑇⟩)
1311, 12eleq2s 2129 . . 3 (𝑋 (𝑆𝐹𝑇) → ⟨𝑆, 𝑇 dom 𝐹)
146, 13sseldi 2937 . 2 (𝑋 (𝑆𝐹𝑇) → ⟨𝑆, 𝑇 (A × B))
15 opelxp 4317 . 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 1242   ∈ wcel 1390  ⟨cop 3370   × cxp 4286  dom cdm 4288  Rel wrel 4293  Fun wfun 4839  ‘cfv 4845  (class class class)co 5455  {coprab 5456   ↦ cmpt2 5457 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460 This theorem is referenced by:  elmpt2cl1  5641  elmpt2cl2  5642  elovmpt2  5643  ixxssxr  8519  elixx3g  8520  ixxssixx  8521  eliooxr  8546  elfz2  8631
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