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Mirrors > Home > MPE Home > Th. List > f1dom | Structured version Visualization version 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 8521 | . 2 ⊢ (𝐵 ∈ V → (𝐹:𝐴–1-1→𝐵 → 𝐴 ≼ 𝐵)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐴 ≼ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2108 Vcvv 3493 class class class wbr 5057 –1-1→wf1 6345 ≼ cdom 8499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-dom 8503 |
This theorem is referenced by: dominf 9859 dominfac 9987 lgsqrlem4 25917 |
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