MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fneqeql2 Structured version   Visualization version   GIF version

Theorem fneqeql2 7080
Description: Two functions are equal iff their equalizer contains the whole domain. (Contributed by Stefan O'Rear, 9-Mar-2015.)
Assertion
Ref Expression
fneqeql2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺𝐴 ⊆ dom (𝐹𝐺)))

Proof of Theorem fneqeql2
StepHypRef Expression
1 fneqeql 7079 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ dom (𝐹𝐺) = 𝐴))
2 eqss 4024 . . 3 (dom (𝐹𝐺) = 𝐴 ↔ (dom (𝐹𝐺) ⊆ 𝐴𝐴 ⊆ dom (𝐹𝐺)))
3 inss1 4258 . . . . . 6 (𝐹𝐺) ⊆ 𝐹
4 dmss 5927 . . . . . 6 ((𝐹𝐺) ⊆ 𝐹 → dom (𝐹𝐺) ⊆ dom 𝐹)
53, 4ax-mp 5 . . . . 5 dom (𝐹𝐺) ⊆ dom 𝐹
6 fndm 6682 . . . . . 6 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
76adantr 480 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom 𝐹 = 𝐴)
85, 7sseqtrid 4061 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) ⊆ 𝐴)
98biantrurd 532 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐴 ⊆ dom (𝐹𝐺) ↔ (dom (𝐹𝐺) ⊆ 𝐴𝐴 ⊆ dom (𝐹𝐺))))
102, 9bitr4id 290 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (dom (𝐹𝐺) = 𝐴𝐴 ⊆ dom (𝐹𝐺)))
111, 10bitrd 279 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺𝐴 ⊆ dom (𝐹𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  cin 3975  wss 3976  dom cdm 5700   Fn wfn 6568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-fv 6581
This theorem is referenced by:  evlseu  22130  hauseqcn  33844
  Copyright terms: Public domain W3C validator