Theorem fneqeql 6992

Description: Two functions are equal iff their equalizer is the whole domain. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Assertion

Ref	Expression
fneqeql	⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ dom (𝐹 ∩ 𝐺) = 𝐴))

Proof of Theorem fneqeql

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqfnfv 6977	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))
2		eqcom 2744	. . . 4 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)} = 𝐴 ↔ 𝐴 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)})
3		rabid2 3423	. . . 4 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)} ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))
4	2, 3	bitri 275	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)} = 𝐴 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))
5	1, 4	bitr4di 289	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)} = 𝐴))
6		fndmin 6991	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∩ 𝐺) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)})
7	6	eqeq1d 2739	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (dom (𝐹 ∩ 𝐺) = 𝐴 ↔ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)} = 𝐴))
8	5, 7	bitr4d 282	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ dom (𝐹 ∩ 𝐺) = 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∀wral 3052  {crab 3390   ∩ cin 3889  dom cdm 5624   Fn wfn 6487  ‘cfv 6492

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500

This theorem is referenced by:  fneqeql2  6993  fnreseql  6994  lspextmo  21043