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Theorem fnreseql 7034
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 6660 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴) → (𝐹𝑋) Fn 𝑋)
213adant2 1131 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴𝑋𝐴) → (𝐹𝑋) Fn 𝑋)
3 fnssres 6660 . . . 4 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐺𝑋) Fn 𝑋)
433adant1 1130 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴𝑋𝐴) → (𝐺𝑋) Fn 𝑋)
5 fneqeql 7032 . . 3 (((𝐹𝑋) Fn 𝑋 ∧ (𝐺𝑋) Fn 𝑋) → ((𝐹𝑋) = (𝐺𝑋) ↔ dom ((𝐹𝑋) ∩ (𝐺𝑋)) = 𝑋))
62, 4, 5syl2anc 584 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝑋) = (𝐺𝑋) ↔ dom ((𝐹𝑋) ∩ (𝐺𝑋)) = 𝑋))
7 resindir 5990 . . . . . 6 ((𝐹𝐺) ↾ 𝑋) = ((𝐹𝑋) ∩ (𝐺𝑋))
87dmeqi 5896 . . . . 5 dom ((𝐹𝐺) ↾ 𝑋) = dom ((𝐹𝑋) ∩ (𝐺𝑋))
9 dmres 5995 . . . . 5 dom ((𝐹𝐺) ↾ 𝑋) = (𝑋 ∩ dom (𝐹𝐺))
108, 9eqtr3i 2761 . . . 4 dom ((𝐹𝑋) ∩ (𝐺𝑋)) = (𝑋 ∩ dom (𝐹𝐺))
1110eqeq1i 2736 . . 3 (dom ((𝐹𝑋) ∩ (𝐺𝑋)) = 𝑋 ↔ (𝑋 ∩ dom (𝐹𝐺)) = 𝑋)
12 df-ss 3961 . . 3 (𝑋 ⊆ dom (𝐹𝐺) ↔ (𝑋 ∩ dom (𝐹𝐺)) = 𝑋)
1311, 12bitr4i 277 . 2 (dom ((𝐹𝑋) ∩ (𝐺𝑋)) = 𝑋𝑋 ⊆ dom (𝐹𝐺))
146, 13bitrdi 286 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝑋) = (𝐺𝑋) ↔ 𝑋 ⊆ dom (𝐹𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1087   = wceq 1541  cin 3943  wss 3944  dom cdm 5669  cres 5671   Fn wfn 6527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-fv 6540
This theorem is referenced by:  lspextmo  20616  evlseu  21575  symgcom2  32116  hauseqcn  32707
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