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Theorem wdomnumr 8925
Description: Weak dominance agrees with normal for numerable right sets. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
Assertion
Ref Expression
wdomnumr (𝐵 ∈ dom card → (𝐴* 𝐵𝐴𝐵))

Proof of Theorem wdomnumr
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 brwdom 8513 . . 3 (𝐵 ∈ dom card → (𝐴* 𝐵 ↔ (𝐴 = ∅ ∨ ∃𝑥 𝑥:𝐵onto𝐴)))
2 0domg 8128 . . . . 5 (𝐵 ∈ dom card → ∅ ≼ 𝐵)
3 breq1 4688 . . . . 5 (𝐴 = ∅ → (𝐴𝐵 ↔ ∅ ≼ 𝐵))
42, 3syl5ibrcom 237 . . . 4 (𝐵 ∈ dom card → (𝐴 = ∅ → 𝐴𝐵))
5 fodomnum 8918 . . . . 5 (𝐵 ∈ dom card → (𝑥:𝐵onto𝐴𝐴𝐵))
65exlimdv 1901 . . . 4 (𝐵 ∈ dom card → (∃𝑥 𝑥:𝐵onto𝐴𝐴𝐵))
74, 6jaod 394 . . 3 (𝐵 ∈ dom card → ((𝐴 = ∅ ∨ ∃𝑥 𝑥:𝐵onto𝐴) → 𝐴𝐵))
81, 7sylbid 230 . 2 (𝐵 ∈ dom card → (𝐴* 𝐵𝐴𝐵))
9 domwdom 8520 . 2 (𝐴𝐵𝐴* 𝐵)
108, 9impbid1 215 1 (𝐵 ∈ dom card → (𝐴* 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382   = wceq 1523  wex 1744  wcel 2030  c0 3948   class class class wbr 4685  dom cdm 5143  ontowfo 5924  cdom 7995  * cwdom 8503  cardccrd 8799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-wdom 8505  df-card 8803  df-acn 8806
This theorem is referenced by:  wdomac  9387  ttac  37920  isnumbasgrplem2  37991
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