Theorem fnreseql 7039

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 𝐴 ∧ 𝐺 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ((𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ↔ 𝑋 ⊆ dom (𝐹 ∩ 𝐺)))

Proof of Theorem fnreseql

Step	Hyp	Ref	Expression
1		fnssres 6654	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝐹 ↾ 𝑋) Fn 𝑋)
2	1	3adant2 1149	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝐹 ↾ 𝑋) Fn 𝑋)
3		fnssres 6654	. . . 4 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝐺 ↾ 𝑋) Fn 𝑋)
4	3	3adant1 1148	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝐺 ↾ 𝑋) Fn 𝑋)
5		fneqeql 7037	. . 3 ⊢ (((𝐹 ↾ 𝑋) Fn 𝑋 ∧ (𝐺 ↾ 𝑋) Fn 𝑋) → ((𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ↔ dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋)) = 𝑋))
6	2, 4, 5	syl2anc 596	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ((𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ↔ dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋)) = 𝑋))
7		resindir 5987	. . . . . 6 ⊢ ((𝐹 ∩ 𝐺) ↾ 𝑋) = ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋))
8	7	dmeqi 5886	. . . . 5 ⊢ dom ((𝐹 ∩ 𝐺) ↾ 𝑋) = dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋))
9		dmres 6003	. . . . 5 ⊢ dom ((𝐹 ∩ 𝐺) ↾ 𝑋) = (𝑋 ∩ dom (𝐹 ∩ 𝐺))
10	8, 9	eqtr3i 2786	. . . 4 ⊢ dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋)) = (𝑋 ∩ dom (𝐹 ∩ 𝐺))
11	10	eqeq1i 2766	. . 3 ⊢ (dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋)) = 𝑋 ↔ (𝑋 ∩ dom (𝐹 ∩ 𝐺)) = 𝑋)
12		dfss2 3917	. . 3 ⊢ (𝑋 ⊆ dom (𝐹 ∩ 𝐺) ↔ (𝑋 ∩ dom (𝐹 ∩ 𝐺)) = 𝑋)
13	11, 12	bitr4i 281	. 2 ⊢ (dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋)) = 𝑋 ↔ 𝑋 ⊆ dom (𝐹 ∩ 𝐺))
14	6, 13	bitrdi 290	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ((𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ↔ 𝑋 ⊆ dom (𝐹 ∩ 𝐺)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∩ cin 3898   ⊆ wss 3899  dom cdm 5651   ↾ cres 5653   Fn wfn 6526

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-fv 6539

This theorem is used by:  lspextmo  21311  evlseu  22372  symgcom2  33627  hauseqcn  34512