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Theorem fneqeql2 7036
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 7035 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ dom (𝐹𝐺) = 𝐴))
2 eqss 3995 . . 3 (dom (𝐹𝐺) = 𝐴 ↔ (dom (𝐹𝐺) ⊆ 𝐴𝐴 ⊆ dom (𝐹𝐺)))
3 inss1 4226 . . . . . 6 (𝐹𝐺) ⊆ 𝐹
4 dmss 5897 . . . . . 6 ((𝐹𝐺) ⊆ 𝐹 → dom (𝐹𝐺) ⊆ dom 𝐹)
53, 4ax-mp 5 . . . . 5 dom (𝐹𝐺) ⊆ dom 𝐹
6 fndm 6644 . . . . . 6 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
76adantr 482 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom 𝐹 = 𝐴)
85, 7sseqtrid 4032 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) ⊆ 𝐴)
98biantrurd 534 . . 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 205  wa 397   = wceq 1542  cin 3945  wss 3946  dom cdm 5672   Fn wfn 6530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fn 6538  df-fv 6543
This theorem is referenced by:  evlseu  21615  hauseqcn  32809
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