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Mirrors > Home > ILE Home > Th. List > f1dom | GIF version |
Description: The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 19-Jun-1998.) |
Ref | Expression |
---|---|
f1dom.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
f1dom | ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐴 ≼ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1dom.1 | . 2 ⊢ 𝐵 ∈ V | |
2 | f1domg 6405 | . 2 ⊢ (𝐵 ∈ V → (𝐹:𝐴–1-1→𝐵 → 𝐴 ≼ 𝐵)) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐴 ≼ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1434 Vcvv 2612 class class class wbr 3811 –1-1→wf1 4966 ≼ cdom 6386 |
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 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3919 ax-sep 3922 ax-pow 3974 ax-pr 4000 ax-un 4224 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-id 4084 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-rn 4412 df-res 4413 df-ima 4414 df-iota 4934 df-fun 4971 df-fn 4972 df-f 4973 df-f1 4974 df-fo 4975 df-f1o 4976 df-fv 4977 df-dom 6389 |
This theorem is referenced by: 1domsn 6465 |
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