Theorem fnreseql 5393

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 5113	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝐹 ↾ 𝑋) Fn 𝑋)
2	1	3adant2 962	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝐹 ↾ 𝑋) Fn 𝑋)
3		fnssres 5113	. . . 4 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝐺 ↾ 𝑋) Fn 𝑋)
4	3	3adant1 961	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝐺 ↾ 𝑋) Fn 𝑋)
5		fneqeql 5391	. . 3 ⊢ (((𝐹 ↾ 𝑋) Fn 𝑋 ∧ (𝐺 ↾ 𝑋) Fn 𝑋) → ((𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ↔ dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋)) = 𝑋))
6	2, 4, 5	syl2anc 403	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ((𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ↔ dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋)) = 𝑋))
7		resindir 4717	. . . . . 6 ⊢ ((𝐹 ∩ 𝐺) ↾ 𝑋) = ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋))
8	7	dmeqi 4625	. . . . 5 ⊢ dom ((𝐹 ∩ 𝐺) ↾ 𝑋) = dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋))
9		dmres 4721	. . . . 5 ⊢ dom ((𝐹 ∩ 𝐺) ↾ 𝑋) = (𝑋 ∩ dom (𝐹 ∩ 𝐺))
10	8, 9	eqtr3i 2110	. . . 4 ⊢ dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋)) = (𝑋 ∩ dom (𝐹 ∩ 𝐺))
11	10	eqeq1i 2095	. . 3 ⊢ (dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋)) = 𝑋 ↔ (𝑋 ∩ dom (𝐹 ∩ 𝐺)) = 𝑋)
12		df-ss 3010	. . 3 ⊢ (𝑋 ⊆ dom (𝐹 ∩ 𝐺) ↔ (𝑋 ∩ dom (𝐹 ∩ 𝐺)) = 𝑋)
13	11, 12	bitr4i 185	. 2 ⊢ (dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋)) = 𝑋 ↔ 𝑋 ⊆ dom (𝐹 ∩ 𝐺))
14	6, 13	syl6bb 194	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ((𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ↔ 𝑋 ⊆ dom (𝐹 ∩ 𝐺)))

Syntax hints:   → wi 4   ↔ wb 103   ∧ w3a 924   = wceq 1289   ∩ cin 2996   ⊆ wss 2997  dom cdm 4428   ↾ cres 4430   Fn wfn 4997

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-res 4440  df-iota 4967  df-fun 5004  df-fn 5005  df-fv 5010

This theorem is referenced by: (None)