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Theorem fnreseql 6868
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 6500 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴) → (𝐹𝑋) Fn 𝑋)
213adant2 1133 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴𝑋𝐴) → (𝐹𝑋) Fn 𝑋)
3 fnssres 6500 . . . 4 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐺𝑋) Fn 𝑋)
433adant1 1132 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴𝑋𝐴) → (𝐺𝑋) Fn 𝑋)
5 fneqeql 6866 . . 3 (((𝐹𝑋) Fn 𝑋 ∧ (𝐺𝑋) Fn 𝑋) → ((𝐹𝑋) = (𝐺𝑋) ↔ dom ((𝐹𝑋) ∩ (𝐺𝑋)) = 𝑋))
62, 4, 5syl2anc 587 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝑋) = (𝐺𝑋) ↔ dom ((𝐹𝑋) ∩ (𝐺𝑋)) = 𝑋))
7 resindir 5868 . . . . . 6 ((𝐹𝐺) ↾ 𝑋) = ((𝐹𝑋) ∩ (𝐺𝑋))
87dmeqi 5773 . . . . 5 dom ((𝐹𝐺) ↾ 𝑋) = dom ((𝐹𝑋) ∩ (𝐺𝑋))
9 dmres 5873 . . . . 5 dom ((𝐹𝐺) ↾ 𝑋) = (𝑋 ∩ dom (𝐹𝐺))
108, 9eqtr3i 2767 . . . 4 dom ((𝐹𝑋) ∩ (𝐺𝑋)) = (𝑋 ∩ dom (𝐹𝐺))
1110eqeq1i 2742 . . 3 (dom ((𝐹𝑋) ∩ (𝐺𝑋)) = 𝑋 ↔ (𝑋 ∩ dom (𝐹𝐺)) = 𝑋)
12 df-ss 3883 . . 3 (𝑋 ⊆ dom (𝐹𝐺) ↔ (𝑋 ∩ dom (𝐹𝐺)) = 𝑋)
1311, 12bitr4i 281 . 2 (dom ((𝐹𝑋) ∩ (𝐺𝑋)) = 𝑋𝑋 ⊆ dom (𝐹𝐺))
146, 13bitrdi 290 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝑋) = (𝐺𝑋) ↔ 𝑋 ⊆ dom (𝐹𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1089   = wceq 1543  cin 3865  wss 3866  dom cdm 5551  cres 5553   Fn wfn 6375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-fv 6388
This theorem is referenced by:  lspextmo  20093  evlseu  21043  symgcom2  31072  hauseqcn  31562
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