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Mirrors > Home > ILE Home > Th. List > dom2d | GIF version |
Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 20-May-2013.) |
Ref | Expression |
---|---|
dom2d.1 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵)) |
dom2d.2 | ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦))) |
Ref | Expression |
---|---|
dom2d | ⊢ (𝜑 → (𝐵 ∈ 𝑅 → 𝐴 ≼ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dom2d.1 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵)) | |
2 | dom2d.2 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦))) | |
3 | 1, 2 | dom2lem 6479 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵) |
4 | f1domg 6465 | . 2 ⊢ (𝐵 ∈ 𝑅 → ((𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵 → 𝐴 ≼ 𝐵)) | |
5 | 3, 4 | syl5com 29 | 1 ⊢ (𝜑 → (𝐵 ∈ 𝑅 → 𝐴 ≼ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1289 ∈ wcel 1438 class class class wbr 3843 ↦ cmpt 3897 –1-1→wf1 5007 ≼ cdom 6446 |
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 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-coll 3952 ax-sep 3955 ax-pow 4007 ax-pr 4034 ax-un 4258 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-reu 2366 df-rab 2368 df-v 2621 df-sbc 2841 df-csb 2934 df-un 3003 df-in 3005 df-ss 3012 df-pw 3429 df-sn 3450 df-pr 3451 df-op 3453 df-uni 3652 df-iun 3730 df-br 3844 df-opab 3898 df-mpt 3899 df-id 4118 df-xp 4442 df-rel 4443 df-cnv 4444 df-co 4445 df-dm 4446 df-rn 4447 df-res 4448 df-ima 4449 df-iota 4975 df-fun 5012 df-fn 5013 df-f 5014 df-f1 5015 df-fo 5016 df-f1o 5017 df-fv 5018 df-dom 6449 |
This theorem is referenced by: dom2 6482 |
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