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Mirrors > Home > ILE Home > Th. List > funbrfv | GIF version |
Description: The second argument of a binary relation on a function is the function's value. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
funbrfv | ⊢ (Fun 𝐹 → (A𝐹B → (𝐹‘A) = B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funrel 4862 | . . . 4 ⊢ (Fun 𝐹 → Rel 𝐹) | |
2 | brrelex2 4326 | . . . 4 ⊢ ((Rel 𝐹 ∧ A𝐹B) → B ∈ V) | |
3 | 1, 2 | sylan 267 | . . 3 ⊢ ((Fun 𝐹 ∧ A𝐹B) → B ∈ V) |
4 | breq2 3759 | . . . . . 6 ⊢ (y = B → (A𝐹y ↔ A𝐹B)) | |
5 | 4 | anbi2d 437 | . . . . 5 ⊢ (y = B → ((Fun 𝐹 ∧ A𝐹y) ↔ (Fun 𝐹 ∧ A𝐹B))) |
6 | eqeq2 2046 | . . . . 5 ⊢ (y = B → ((𝐹‘A) = y ↔ (𝐹‘A) = B)) | |
7 | 5, 6 | imbi12d 223 | . . . 4 ⊢ (y = B → (((Fun 𝐹 ∧ A𝐹y) → (𝐹‘A) = y) ↔ ((Fun 𝐹 ∧ A𝐹B) → (𝐹‘A) = B))) |
8 | funeu 4869 | . . . . . 6 ⊢ ((Fun 𝐹 ∧ A𝐹y) → ∃!y A𝐹y) | |
9 | tz6.12-1 5143 | . . . . . 6 ⊢ ((A𝐹y ∧ ∃!y A𝐹y) → (𝐹‘A) = y) | |
10 | 8, 9 | sylan2 270 | . . . . 5 ⊢ ((A𝐹y ∧ (Fun 𝐹 ∧ A𝐹y)) → (𝐹‘A) = y) |
11 | 10 | anabss7 517 | . . . 4 ⊢ ((Fun 𝐹 ∧ A𝐹y) → (𝐹‘A) = y) |
12 | 7, 11 | vtoclg 2607 | . . 3 ⊢ (B ∈ V → ((Fun 𝐹 ∧ A𝐹B) → (𝐹‘A) = B)) |
13 | 3, 12 | mpcom 32 | . 2 ⊢ ((Fun 𝐹 ∧ A𝐹B) → (𝐹‘A) = B) |
14 | 13 | ex 108 | 1 ⊢ (Fun 𝐹 → (A𝐹B → (𝐹‘A) = B)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1242 ∈ wcel 1390 ∃!weu 1897 Vcvv 2551 class class class wbr 3755 Rel wrel 4293 Fun wfun 4839 ‘cfv 4845 |
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-v 2553 df-sbc 2759 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-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-iota 4810 df-fun 4847 df-fv 4853 |
This theorem is referenced by: funopfv 5156 fnbrfvb 5157 fvelima 5168 fvi 5173 fmptco 5273 fliftfun 5379 fliftval 5383 tfrlem5 5871 |
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