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Mirrors > Home > ILE Home > Th. List > mpofun | GIF version |
Description: The maps-to notation for an operation is always a function. (Contributed by Scott Fenton, 21-Mar-2012.) |
Ref | Expression |
---|---|
mpofun.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
Ref | Expression |
---|---|
mpofun | ⊢ Fun 𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqtr3 2159 | . . . . . 6 ⊢ ((𝑧 = 𝐶 ∧ 𝑤 = 𝐶) → 𝑧 = 𝑤) | |
2 | 1 | ad2ant2l 499 | . . . . 5 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)) → 𝑧 = 𝑤) |
3 | 2 | gen2 1426 | . . . 4 ⊢ ∀𝑧∀𝑤((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)) → 𝑧 = 𝑤) |
4 | eqeq1 2146 | . . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑧 = 𝐶 ↔ 𝑤 = 𝐶)) | |
5 | 4 | anbi2d 459 | . . . . 5 ⊢ (𝑧 = 𝑤 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶))) |
6 | 5 | mo4 2060 | . . . 4 ⊢ (∃*𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ∀𝑧∀𝑤((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)) → 𝑧 = 𝑤)) |
7 | 3, 6 | mpbir 145 | . . 3 ⊢ ∃*𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) |
8 | 7 | funoprab 5871 | . 2 ⊢ Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
9 | mpofun.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
10 | df-mpo 5779 | . . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} | |
11 | 9, 10 | eqtri 2160 | . . 3 ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
12 | 11 | funeqi 5144 | . 2 ⊢ (Fun 𝐹 ↔ Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}) |
13 | 8, 12 | mpbir 145 | 1 ⊢ Fun 𝐹 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1329 = wceq 1331 ∈ wcel 1480 ∃*wmo 2000 Fun wfun 5117 {coprab 5775 ∈ cmpo 5776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-fun 5125 df-oprab 5778 df-mpo 5779 |
This theorem is referenced by: elmpocl 5968 ofexg 5986 mpoexxg 6108 mpoxopn0yelv 6136 |
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