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Mirrors > Home > MPE Home > Th. List > dom2d | Structured version Visualization version 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 8149 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵) |
4 | f1domg 8129 | . 2 ⊢ (𝐵 ∈ 𝑅 → ((𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵 → 𝐴 ≼ 𝐵)) | |
5 | 3, 4 | syl5com 31 | 1 ⊢ (𝜑 → (𝐵 ∈ 𝑅 → 𝐴 ≼ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 class class class wbr 4786 ↦ cmpt 4863 –1-1→wf1 6026 ≼ cdom 8107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-dom 8111 |
This theorem is referenced by: dom2 8152 fineqvlem 8330 fseqdom 9049 fin1a2lem9 9432 iundom2g 9564 canthwe 9675 prmreclem2 15824 prmreclem3 15825 sylow1lem4 18219 aannenlem1 24299 derangenlem 31487 fphpd 37903 pellexlem3 37918 unxpwdom3 38188 |
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