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Theorem fodom 9659
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.)
Hypothesis
Ref Expression
fodom.1 𝐴 ∈ V
Assertion
Ref Expression
fodom (𝐹:𝐴onto𝐵𝐵𝐴)

Proof of Theorem fodom
StepHypRef Expression
1 fodom.1 . 2 𝐴 ∈ V
2 numth3 9607 . 2 (𝐴 ∈ V → 𝐴 ∈ dom card)
3 fodomnum 9193 . 2 (𝐴 ∈ dom card → (𝐹:𝐴onto𝐵𝐵𝐴))
41, 2, 3mp2b 10 1 (𝐹:𝐴onto𝐵𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2166  Vcvv 3414   class class class wbr 4873  dom cdm 5342  ontowfo 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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