MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnreseql Structured version   Visualization version   GIF version

Theorem fnreseql 7067
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 6691 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴) → (𝐹𝑋) Fn 𝑋)
213adant2 1130 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴𝑋𝐴) → (𝐹𝑋) Fn 𝑋)
3 fnssres 6691 . . . 4 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐺𝑋) Fn 𝑋)
433adant1 1129 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴𝑋𝐴) → (𝐺𝑋) Fn 𝑋)
5 fneqeql 7065 . . 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 6016 . . . . . 6 ((𝐹𝐺) ↾ 𝑋) = ((𝐹𝑋) ∩ (𝐺𝑋))
87dmeqi 5917 . . . . 5 dom ((𝐹𝐺) ↾ 𝑋) = dom ((𝐹𝑋) ∩ (𝐺𝑋))
9 dmres 6031 . . . . 5 dom ((𝐹𝐺) ↾ 𝑋) = (𝑋 ∩ dom (𝐹𝐺))
108, 9eqtr3i 2764 . . . 4 dom ((𝐹𝑋) ∩ (𝐺𝑋)) = (𝑋 ∩ dom (𝐹𝐺))
1110eqeq1i 2739 . . 3 (dom ((𝐹𝑋) ∩ (𝐺𝑋)) = 𝑋 ↔ (𝑋 ∩ dom (𝐹𝐺)) = 𝑋)
12 dfss2 3980 . . 3 (𝑋 ⊆ dom (𝐹𝐺) ↔ (𝑋 ∩ dom (𝐹𝐺)) = 𝑋)
1311, 12bitr4i 278 . 2 (dom ((𝐹𝑋) ∩ (𝐺𝑋)) = 𝑋𝑋 ⊆ dom (𝐹𝐺))
146, 13bitrdi 287 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝑋) = (𝐺𝑋) ↔ 𝑋 ⊆ dom (𝐹𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1086   = wceq 1536  cin 3961  wss 3962  dom cdm 5688  cres 5690   Fn wfn 6557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-fv 6570
This theorem is referenced by:  lspextmo  21072  evlseu  22124  symgcom2  33086  hauseqcn  33858
  Copyright terms: Public domain W3C validator