Theorem fnreseql 6907

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 6539	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝐹 ↾ 𝑋) Fn 𝑋)
2	1	3adant2 1129	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝐹 ↾ 𝑋) Fn 𝑋)
3		fnssres 6539	. . . 4 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝐺 ↾ 𝑋) Fn 𝑋)
4	3	3adant1 1128	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝐺 ↾ 𝑋) Fn 𝑋)
5		fneqeql 6905	. . 3 ⊢ (((𝐹 ↾ 𝑋) Fn 𝑋 ∧ (𝐺 ↾ 𝑋) Fn 𝑋) → ((𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ↔ dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋)) = 𝑋))
6	2, 4, 5	syl2anc 583	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ((𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ↔ dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋)) = 𝑋))
7		resindir 5897	. . . . . 6 ⊢ ((𝐹 ∩ 𝐺) ↾ 𝑋) = ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋))
8	7	dmeqi 5802	. . . . 5 ⊢ dom ((𝐹 ∩ 𝐺) ↾ 𝑋) = dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋))
9		dmres 5902	. . . . 5 ⊢ dom ((𝐹 ∩ 𝐺) ↾ 𝑋) = (𝑋 ∩ dom (𝐹 ∩ 𝐺))
10	8, 9	eqtr3i 2768	. . . 4 ⊢ dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋)) = (𝑋 ∩ dom (𝐹 ∩ 𝐺))
11	10	eqeq1i 2743	. . 3 ⊢ (dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋)) = 𝑋 ↔ (𝑋 ∩ dom (𝐹 ∩ 𝐺)) = 𝑋)
12		df-ss 3900	. . 3 ⊢ (𝑋 ⊆ dom (𝐹 ∩ 𝐺) ↔ (𝑋 ∩ dom (𝐹 ∩ 𝐺)) = 𝑋)
13	11, 12	bitr4i 277	. 2 ⊢ (dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋)) = 𝑋 ↔ 𝑋 ⊆ dom (𝐹 ∩ 𝐺))
14	6, 13	bitrdi 286	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ((𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ↔ 𝑋 ⊆ dom (𝐹 ∩ 𝐺)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1085   = wceq 1539   ∩ cin 3882   ⊆ wss 3883  dom cdm 5580   ↾ cres 5582   Fn wfn 6413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-fv 6426

This theorem is referenced by:  lspextmo  20233  evlseu  21203  symgcom2  31255  hauseqcn  31750