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Theorem funopab4 4880
 Description: A class of ordered pairs of values in the form used by df-mpt 3811 is a function. (Contributed by NM, 17-Feb-2013.)
Assertion
Ref Expression
funopab4 Fun {⟨x, y⟩ ∣ (φ y = A)}
Distinct variable groups:   x,y   y,A
Allowed substitution hints:   φ(x,y)   A(x)

Proof of Theorem funopab4
StepHypRef Expression
1 simpr 103 . . 3 ((φ y = A) → y = A)
21ssopab2i 4005 . 2 {⟨x, y⟩ ∣ (φ y = A)} ⊆ {⟨x, y⟩ ∣ y = A}
3 funopabeq 4879 . 2 Fun {⟨x, y⟩ ∣ y = A}
4 funss 4863 . 2 ({⟨x, y⟩ ∣ (φ y = A)} ⊆ {⟨x, y⟩ ∣ y = A} → (Fun {⟨x, y⟩ ∣ y = A} → Fun {⟨x, y⟩ ∣ (φ y = A)}))
52, 3, 4mp2 16 1 Fun {⟨x, y⟩ ∣ (φ y = A)}
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1242   ⊆ wss 2911  {copab 3808  Fun wfun 4839 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-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-fun 4847 This theorem is referenced by:  funmpt  4881
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