ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fneqeql2 GIF version

Theorem fneqeql2 5304
Description: Two functions are equal iff their equalizer contains the whole domain. (Contributed by Stefan O'Rear, 9-Mar-2015.)
Assertion
Ref Expression
fneqeql2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺𝐴 ⊆ dom (𝐹𝐺)))

Proof of Theorem fneqeql2
StepHypRef Expression
1 fneqeql 5303 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ dom (𝐹𝐺) = 𝐴))
2 inss1 3185 . . . . . 6 (𝐹𝐺) ⊆ 𝐹
3 dmss 4562 . . . . . 6 ((𝐹𝐺) ⊆ 𝐹 → dom (𝐹𝐺) ⊆ dom 𝐹)
42, 3ax-mp 7 . . . . 5 dom (𝐹𝐺) ⊆ dom 𝐹
5 fndm 5026 . . . . . 6 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
65adantr 265 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom 𝐹 = 𝐴)
74, 6syl5sseq 3021 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) ⊆ 𝐴)
87biantrurd 293 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐴 ⊆ dom (𝐹𝐺) ↔ (dom (𝐹𝐺) ⊆ 𝐴𝐴 ⊆ dom (𝐹𝐺))))
9 eqss 2988 . . 3 (dom (𝐹𝐺) = 𝐴 ↔ (dom (𝐹𝐺) ⊆ 𝐴𝐴 ⊆ dom (𝐹𝐺)))
108, 9syl6rbbr 192 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (dom (𝐹𝐺) = 𝐴𝐴 ⊆ dom (𝐹𝐺)))
111, 10bitrd 181 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺𝐴 ⊆ dom (𝐹𝐺)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  cin 2944  wss 2945  dom cdm 4373   Fn wfn 4925
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fn 4933  df-fv 4938
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator