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Theorem fnreseql 6283
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 5962 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴) → (𝐹𝑋) Fn 𝑋)
213adant2 1078 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴𝑋𝐴) → (𝐹𝑋) Fn 𝑋)
3 fnssres 5962 . . . 4 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐺𝑋) Fn 𝑋)
433adant1 1077 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴𝑋𝐴) → (𝐺𝑋) Fn 𝑋)
5 fneqeql 6281 . . 3 (((𝐹𝑋) Fn 𝑋 ∧ (𝐺𝑋) Fn 𝑋) → ((𝐹𝑋) = (𝐺𝑋) ↔ dom ((𝐹𝑋) ∩ (𝐺𝑋)) = 𝑋))
62, 4, 5syl2anc 692 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝑋) = (𝐺𝑋) ↔ dom ((𝐹𝑋) ∩ (𝐺𝑋)) = 𝑋))
7 resindir 5372 . . . . . 6 ((𝐹𝐺) ↾ 𝑋) = ((𝐹𝑋) ∩ (𝐺𝑋))
87dmeqi 5285 . . . . 5 dom ((𝐹𝐺) ↾ 𝑋) = dom ((𝐹𝑋) ∩ (𝐺𝑋))
9 dmres 5378 . . . . 5 dom ((𝐹𝐺) ↾ 𝑋) = (𝑋 ∩ dom (𝐹𝐺))
108, 9eqtr3i 2645 . . . 4 dom ((𝐹𝑋) ∩ (𝐺𝑋)) = (𝑋 ∩ dom (𝐹𝐺))
1110eqeq1i 2626 . . 3 (dom ((𝐹𝑋) ∩ (𝐺𝑋)) = 𝑋 ↔ (𝑋 ∩ dom (𝐹𝐺)) = 𝑋)
12 df-ss 3569 . . 3 (𝑋 ⊆ dom (𝐹𝐺) ↔ (𝑋 ∩ dom (𝐹𝐺)) = 𝑋)
1311, 12bitr4i 267 . 2 (dom ((𝐹𝑋) ∩ (𝐺𝑋)) = 𝑋𝑋 ⊆ dom (𝐹𝐺))
146, 13syl6bb 276 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝑋) = (𝐺𝑋) ↔ 𝑋 ⊆ dom (𝐹𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1036   = wceq 1480  cin 3554  wss 3555  dom cdm 5074  cres 5076   Fn wfn 5842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-fv 5855
This theorem is referenced by:  lspextmo  18975  evlseu  19435  hauseqcn  29723
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