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Theorem fneqeql2 5201
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 A 𝐺 Fn A) → (𝐹 = 𝐺A ⊆ dom (𝐹𝐺)))

Proof of Theorem fneqeql2
StepHypRef Expression
1 fneqeql 5200 . 2 ((𝐹 Fn A 𝐺 Fn A) → (𝐹 = 𝐺 ↔ dom (𝐹𝐺) = A))
2 inss1 3134 . . . . . 6 (𝐹𝐺) ⊆ 𝐹
3 dmss 4461 . . . . . 6 ((𝐹𝐺) ⊆ 𝐹 → dom (𝐹𝐺) ⊆ dom 𝐹)
42, 3ax-mp 7 . . . . 5 dom (𝐹𝐺) ⊆ dom 𝐹
5 fndm 4924 . . . . . 6 (𝐹 Fn A → dom 𝐹 = A)
65adantr 261 . . . . 5 ((𝐹 Fn A 𝐺 Fn A) → dom 𝐹 = A)
74, 6syl5sseq 2970 . . . 4 ((𝐹 Fn A 𝐺 Fn A) → dom (𝐹𝐺) ⊆ A)
87biantrurd 289 . . 3 ((𝐹 Fn A 𝐺 Fn A) → (A ⊆ dom (𝐹𝐺) ↔ (dom (𝐹𝐺) ⊆ A A ⊆ dom (𝐹𝐺))))
9 eqss 2937 . . 3 (dom (𝐹𝐺) = A ↔ (dom (𝐹𝐺) ⊆ A A ⊆ dom (𝐹𝐺)))
108, 9syl6rbbr 188 . 2 ((𝐹 Fn A 𝐺 Fn A) → (dom (𝐹𝐺) = AA ⊆ dom (𝐹𝐺)))
111, 10bitrd 177 1 ((𝐹 Fn A 𝐺 Fn A) → (𝐹 = 𝐺A ⊆ dom (𝐹𝐺)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228  cin 2893  wss 2894  dom cdm 4272   Fn wfn 4824
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-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  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-mpt 3794  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-fn 4832  df-fv 4837
This theorem is referenced by: (None)
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