Theorem fnreseql 7002

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 6623	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝐹 ↾ 𝑋) Fn 𝑋)
2	1	3adant2 1132	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝐹 ↾ 𝑋) Fn 𝑋)
3		fnssres 6623	. . . 4 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝐺 ↾ 𝑋) Fn 𝑋)
4	3	3adant1 1131	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝐺 ↾ 𝑋) Fn 𝑋)
5		fneqeql 7000	. . 3 ⊢ (((𝐹 ↾ 𝑋) Fn 𝑋 ∧ (𝐺 ↾ 𝑋) Fn 𝑋) → ((𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ↔ dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋)) = 𝑋))
6	2, 4, 5	syl2anc 585	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ((𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ↔ dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋)) = 𝑋))
7		resindir 5963	. . . . . 6 ⊢ ((𝐹 ∩ 𝐺) ↾ 𝑋) = ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋))
8	7	dmeqi 5861	. . . . 5 ⊢ dom ((𝐹 ∩ 𝐺) ↾ 𝑋) = dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋))
9		dmres 5979	. . . . 5 ⊢ dom ((𝐹 ∩ 𝐺) ↾ 𝑋) = (𝑋 ∩ dom (𝐹 ∩ 𝐺))
10	8, 9	eqtr3i 2762	. . . 4 ⊢ dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋)) = (𝑋 ∩ dom (𝐹 ∩ 𝐺))
11	10	eqeq1i 2742	. . 3 ⊢ (dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋)) = 𝑋 ↔ (𝑋 ∩ dom (𝐹 ∩ 𝐺)) = 𝑋)
12		dfss2 3921	. . 3 ⊢ (𝑋 ⊆ dom (𝐹 ∩ 𝐺) ↔ (𝑋 ∩ dom (𝐹 ∩ 𝐺)) = 𝑋)
13	11, 12	bitr4i 278	. 2 ⊢ (dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋)) = 𝑋 ↔ 𝑋 ⊆ dom (𝐹 ∩ 𝐺))
14	6, 13	bitrdi 287	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ((𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ↔ 𝑋 ⊆ dom (𝐹 ∩ 𝐺)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1087   = wceq 1542   ∩ cin 3902   ⊆ wss 3903  dom cdm 5632   ↾ cres 5634   Fn wfn 6495

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 5243  ax-nul 5253  ax-pr 5379

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 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-fv 6508

This theorem is referenced by:  lspextmo  21020  evlseu  22050  symgcom2  33177  hauseqcn  34075