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Theorem fnreseql 5329
Description: Two functions are equal on a subset iff their equalizer contains that subset. (Contributed by Stefan O'Rear, 7-Mar-2015.)
Assertion
Ref Expression
fnreseql ((𝐹 Fn 𝐴𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝑋) = (𝐺𝑋) ↔ 𝑋 ⊆ dom (𝐹𝐺)))

Proof of Theorem fnreseql
StepHypRef Expression
1 fnssres 5063 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴) → (𝐹𝑋) Fn 𝑋)
213adant2 958 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴𝑋𝐴) → (𝐹𝑋) Fn 𝑋)
3 fnssres 5063 . . . 4 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐺𝑋) Fn 𝑋)
433adant1 957 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴𝑋𝐴) → (𝐺𝑋) Fn 𝑋)
5 fneqeql 5327 . . 3 (((𝐹𝑋) Fn 𝑋 ∧ (𝐺𝑋) Fn 𝑋) → ((𝐹𝑋) = (𝐺𝑋) ↔ dom ((𝐹𝑋) ∩ (𝐺𝑋)) = 𝑋))
62, 4, 5syl2anc 403 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝑋) = (𝐺𝑋) ↔ dom ((𝐹𝑋) ∩ (𝐺𝑋)) = 𝑋))
7 resindir 4676 . . . . . 6 ((𝐹𝐺) ↾ 𝑋) = ((𝐹𝑋) ∩ (𝐺𝑋))
87dmeqi 4584 . . . . 5 dom ((𝐹𝐺) ↾ 𝑋) = dom ((𝐹𝑋) ∩ (𝐺𝑋))
9 dmres 4680 . . . . 5 dom ((𝐹𝐺) ↾ 𝑋) = (𝑋 ∩ dom (𝐹𝐺))
108, 9eqtr3i 2105 . . . 4 dom ((𝐹𝑋) ∩ (𝐺𝑋)) = (𝑋 ∩ dom (𝐹𝐺))
1110eqeq1i 2090 . . 3 (dom ((𝐹𝑋) ∩ (𝐺𝑋)) = 𝑋 ↔ (𝑋 ∩ dom (𝐹𝐺)) = 𝑋)
12 df-ss 2995 . . 3 (𝑋 ⊆ dom (𝐹𝐺) ↔ (𝑋 ∩ dom (𝐹𝐺)) = 𝑋)
1311, 12bitr4i 185 . 2 (dom ((𝐹𝑋) ∩ (𝐺𝑋)) = 𝑋𝑋 ⊆ dom (𝐹𝐺))
146, 13syl6bb 194 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝑋) = (𝐺𝑋) ↔ 𝑋 ⊆ dom (𝐹𝐺)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  w3a 920   = wceq 1285  cin 2981  wss 2982  dom cdm 4391  cres 4393   Fn wfn 4947
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-res 4403  df-iota 4917  df-fun 4954  df-fn 4955  df-fv 4960
This theorem is referenced by: (None)
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