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Mirrors > Home > ILE Home > Th. List > fnreseql | GIF version |
Description: Two functions are equal on a subset iff their equalizer contains that subset. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
Ref | Expression |
---|---|
fnreseql | ⊢ ((𝐹 Fn A ∧ 𝐺 Fn A ∧ 𝑋 ⊆ A) → ((𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ↔ 𝑋 ⊆ dom (𝐹 ∩ 𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnssres 4955 | . . . 4 ⊢ ((𝐹 Fn A ∧ 𝑋 ⊆ A) → (𝐹 ↾ 𝑋) Fn 𝑋) | |
2 | 1 | 3adant2 922 | . . 3 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn A ∧ 𝑋 ⊆ A) → (𝐹 ↾ 𝑋) Fn 𝑋) |
3 | fnssres 4955 | . . . 4 ⊢ ((𝐺 Fn A ∧ 𝑋 ⊆ A) → (𝐺 ↾ 𝑋) Fn 𝑋) | |
4 | 3 | 3adant1 921 | . . 3 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn A ∧ 𝑋 ⊆ A) → (𝐺 ↾ 𝑋) Fn 𝑋) |
5 | fneqeql 5218 | . . 3 ⊢ (((𝐹 ↾ 𝑋) Fn 𝑋 ∧ (𝐺 ↾ 𝑋) Fn 𝑋) → ((𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ↔ dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋)) = 𝑋)) | |
6 | 2, 4, 5 | syl2anc 391 | . 2 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn A ∧ 𝑋 ⊆ A) → ((𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ↔ dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋)) = 𝑋)) |
7 | resindir 4571 | . . . . . 6 ⊢ ((𝐹 ∩ 𝐺) ↾ 𝑋) = ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋)) | |
8 | 7 | dmeqi 4479 | . . . . 5 ⊢ dom ((𝐹 ∩ 𝐺) ↾ 𝑋) = dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋)) |
9 | dmres 4575 | . . . . 5 ⊢ dom ((𝐹 ∩ 𝐺) ↾ 𝑋) = (𝑋 ∩ dom (𝐹 ∩ 𝐺)) | |
10 | 8, 9 | eqtr3i 2059 | . . . 4 ⊢ dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋)) = (𝑋 ∩ dom (𝐹 ∩ 𝐺)) |
11 | 10 | eqeq1i 2044 | . . 3 ⊢ (dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋)) = 𝑋 ↔ (𝑋 ∩ dom (𝐹 ∩ 𝐺)) = 𝑋) |
12 | df-ss 2925 | . . 3 ⊢ (𝑋 ⊆ dom (𝐹 ∩ 𝐺) ↔ (𝑋 ∩ dom (𝐹 ∩ 𝐺)) = 𝑋) | |
13 | 11, 12 | bitr4i 176 | . 2 ⊢ (dom ((𝐹 ↾ 𝑋) ∩ (𝐺 ↾ 𝑋)) = 𝑋 ↔ 𝑋 ⊆ dom (𝐹 ∩ 𝐺)) |
14 | 6, 13 | syl6bb 185 | 1 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn A ∧ 𝑋 ⊆ A) → ((𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ↔ 𝑋 ⊆ dom (𝐹 ∩ 𝐺))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∧ w3a 884 = wceq 1242 ∩ cin 2910 ⊆ wss 2911 dom cdm 4288 ↾ cres 4290 Fn wfn 4840 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-res 4300 df-iota 4810 df-fun 4847 df-fn 4848 df-fv 4853 |
This theorem is referenced by: (None) |
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