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Mirrors > Home > MPE Home > Th. List > fodom | Structured version Visualization version GIF version |
Description: An onto function implies dominance of domain over range. Lemma 10.20 of [Kunen] p. 30. This theorem uses the Axiom of Choice ac7g 9611. AC is not needed for finite sets - see fodomfi 8508. See also fodomnum 9193. (Contributed by NM, 23-Jul-2004.) |
Ref | Expression |
---|---|
fodom.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
fodom | ⊢ (𝐹:𝐴–onto→𝐵 → 𝐵 ≼ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fodom.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | numth3 9607 | . 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ dom card) | |
3 | fodomnum 9193 | . 2 ⊢ (𝐴 ∈ dom card → (𝐹:𝐴–onto→𝐵 → 𝐵 ≼ 𝐴)) | |
4 | 1, 2, 3 | mp2b 10 | 1 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐵 ≼ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2166 Vcvv 3414 class class class wbr 4873 dom cdm 5342 –onto→wfo 6121 ≼ cdom 8220 cardccrd 9074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-ac2 9600 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-er 8009 df-map 8124 df-en 8223 df-dom 8224 df-card 9078 df-acn 9081 df-ac 9252 |
This theorem is referenced by: fodomg 9660 brdom3 9665 brdom5 9666 brdom4 9667 |
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