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Theorem elmpt2cl2 5642
Description: If a two-parameter class is not empty, the second argument is in its nominal domain. (Contributed by FL, 15-Oct-2012.) (Revised by Stefan O'Rear, 7-Mar-2015.)
Hypothesis
Ref Expression
elmpt2cl.f 𝐹 = (x A, y B𝐶)
Assertion
Ref Expression
elmpt2cl2 (𝑋 (𝑆𝐹𝑇) → 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 elmpt2cl2
StepHypRef Expression
1 elmpt2cl.f . . 3 𝐹 = (x A, y B𝐶)
21elmpt2cl 5640 . 2 (𝑋 (𝑆𝐹𝑇) → (𝑆 A 𝑇 B))
32simprd 107 1 (𝑋 (𝑆𝐹𝑇) → 𝑇 B)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  (class class class)co 5455  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-bndl 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:  iccssico2  8586  elfzoel2  8773
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