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Mirrors > Home > ILE Home > Th. List > dff4im | GIF version |
Description: Property of a mapping. (Contributed by Jim Kingdon, 4-Jan-2019.) |
Ref | Expression |
---|---|
dff4im | ⊢ (𝐹:A⟶B → (𝐹 ⊆ (A × B) ∧ ∀x ∈ A ∃!y ∈ B x𝐹y)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dff3im 5255 | . 2 ⊢ (𝐹:A⟶B → (𝐹 ⊆ (A × B) ∧ ∀x ∈ A ∃!y x𝐹y)) | |
2 | df-br 3756 | . . . . . . . 8 ⊢ (x𝐹y ↔ 〈x, y〉 ∈ 𝐹) | |
3 | ssel 2933 | . . . . . . . . 9 ⊢ (𝐹 ⊆ (A × B) → (〈x, y〉 ∈ 𝐹 → 〈x, y〉 ∈ (A × B))) | |
4 | opelxp2 4321 | . . . . . . . . 9 ⊢ (〈x, y〉 ∈ (A × B) → y ∈ B) | |
5 | 3, 4 | syl6 29 | . . . . . . . 8 ⊢ (𝐹 ⊆ (A × B) → (〈x, y〉 ∈ 𝐹 → y ∈ B)) |
6 | 2, 5 | syl5bi 141 | . . . . . . 7 ⊢ (𝐹 ⊆ (A × B) → (x𝐹y → y ∈ B)) |
7 | 6 | pm4.71rd 374 | . . . . . 6 ⊢ (𝐹 ⊆ (A × B) → (x𝐹y ↔ (y ∈ B ∧ x𝐹y))) |
8 | 7 | eubidv 1905 | . . . . 5 ⊢ (𝐹 ⊆ (A × B) → (∃!y x𝐹y ↔ ∃!y(y ∈ B ∧ x𝐹y))) |
9 | df-reu 2307 | . . . . 5 ⊢ (∃!y ∈ B x𝐹y ↔ ∃!y(y ∈ B ∧ x𝐹y)) | |
10 | 8, 9 | syl6bbr 187 | . . . 4 ⊢ (𝐹 ⊆ (A × B) → (∃!y x𝐹y ↔ ∃!y ∈ B x𝐹y)) |
11 | 10 | ralbidv 2320 | . . 3 ⊢ (𝐹 ⊆ (A × B) → (∀x ∈ A ∃!y x𝐹y ↔ ∀x ∈ A ∃!y ∈ B x𝐹y)) |
12 | 11 | pm5.32i 427 | . 2 ⊢ ((𝐹 ⊆ (A × B) ∧ ∀x ∈ A ∃!y x𝐹y) ↔ (𝐹 ⊆ (A × B) ∧ ∀x ∈ A ∃!y ∈ B x𝐹y)) |
13 | 1, 12 | sylib 127 | 1 ⊢ (𝐹:A⟶B → (𝐹 ⊆ (A × B) ∧ ∀x ∈ A ∃!y ∈ B x𝐹y)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1390 ∃!weu 1897 ∀wral 2300 ∃!wreu 2302 ⊆ wss 2911 〈cop 3370 class class class wbr 3755 × cxp 4286 ⟶wf 4841 |
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-reu 2307 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-rn 4299 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-fv 4853 |
This theorem is referenced by: (None) |
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