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Mirrors > Home > MPE Home > Th. List > fnrndomg | Structured version Visualization version GIF version |
Description: The range of a function is dominated by its domain. (Contributed by NM, 1-Sep-2004.) |
Ref | Expression |
---|---|
fnrndomg | ⊢ (𝐴 ∈ 𝐵 → (𝐹 Fn 𝐴 → ran 𝐹 ≼ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffn4 6464 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹) | |
2 | fodomg 9791 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐹:𝐴–onto→ran 𝐹 → ran 𝐹 ≼ 𝐴)) | |
3 | 1, 2 | syl5bi 243 | 1 ⊢ (𝐴 ∈ 𝐵 → (𝐹 Fn 𝐴 → ran 𝐹 ≼ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2081 class class class wbr 4962 ran crn 5444 Fn wfn 6220 –onto→wfo 6223 ≼ cdom 8355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-ac2 9731 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-se 5403 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-isom 6234 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-er 8139 df-map 8258 df-en 8358 df-dom 8359 df-card 9214 df-acn 9217 df-ac 9388 |
This theorem is referenced by: fnct 9805 unirnfdomd 9835 konigthlem 9836 abrexdomjm 29959 ffsrn 30153 abrexdom 34537 indexdom 34541 subsaliuncl 42183 omeiunle 42341 smflimlem6 42594 |
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