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Theorem wdomnumr 9489
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 9030 . . 3 (𝐵 ∈ dom card → (𝐴* 𝐵 ↔ (𝐴 = ∅ ∨ ∃𝑥 𝑥:𝐵onto𝐴)))
2 0domg 8643 . . . . 5 (𝐵 ∈ dom card → ∅ ≼ 𝐵)
3 breq1 5068 . . . . 5 (𝐴 = ∅ → (𝐴𝐵 ↔ ∅ ≼ 𝐵))
42, 3syl5ibrcom 249 . . . 4 (𝐵 ∈ dom card → (𝐴 = ∅ → 𝐴𝐵))
5 fodomnum 9482 . . . . 5 (𝐵 ∈ dom card → (𝑥:𝐵onto𝐴𝐴𝐵))
65exlimdv 1930 . . . 4 (𝐵 ∈ dom card → (∃𝑥 𝑥:𝐵onto𝐴𝐴𝐵))
74, 6jaod 855 . . 3 (𝐵 ∈ dom card → ((𝐴 = ∅ ∨ ∃𝑥 𝑥:𝐵onto𝐴) → 𝐴𝐵))
81, 7sylbid 242 . 2 (𝐵 ∈ dom card → (𝐴* 𝐵𝐴𝐵))
9 domwdom 9037 . 2 (𝐴𝐵𝐴* 𝐵)
108, 9impbid1 227 1 (𝐵 ∈ dom card → (𝐴* 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wo 843   = wceq 1533  wex 1776  wcel 2110  c0 4290   class class class wbr 5065  dom cdm 5554  ontowfo 6352  cdom 8506  * cwdom 9020  cardccrd 9363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-er 8288  df-map 8407  df-en 8509  df-dom 8510  df-sdom 8511  df-wdom 9022  df-card 9367  df-acn 9370
This theorem is referenced by:  wdomac  9948  ttac  39633  isnumbasgrplem2  39704
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