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Theorem wdomnumr 9174
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 8715 . . 3 (𝐵 ∈ dom card → (𝐴* 𝐵 ↔ (𝐴 = ∅ ∨ ∃𝑥 𝑥:𝐵onto𝐴)))
2 0domg 8330 . . . . 5 (𝐵 ∈ dom card → ∅ ≼ 𝐵)
3 breq1 4847 . . . . 5 (𝐴 = ∅ → (𝐴𝐵 ↔ ∅ ≼ 𝐵))
42, 3syl5ibrcom 239 . . . 4 (𝐵 ∈ dom card → (𝐴 = ∅ → 𝐴𝐵))
5 fodomnum 9167 . . . . 5 (𝐵 ∈ dom card → (𝑥:𝐵onto𝐴𝐴𝐵))
65exlimdv 2029 . . . 4 (𝐵 ∈ dom card → (∃𝑥 𝑥:𝐵onto𝐴𝐴𝐵))
74, 6jaod 886 . . 3 (𝐵 ∈ dom card → ((𝐴 = ∅ ∨ ∃𝑥 𝑥:𝐵onto𝐴) → 𝐴𝐵))
81, 7sylbid 232 . 2 (𝐵 ∈ dom card → (𝐴* 𝐵𝐴𝐵))
9 domwdom 8722 . 2 (𝐴𝐵𝐴* 𝐵)
108, 9impbid1 217 1 (𝐵 ∈ dom card → (𝐴* 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wo 874   = wceq 1653  wex 1875  wcel 2157  c0 4116   class class class wbr 4844  dom cdm 5313  ontowfo 6100  cdom 8194  * cwdom 8705  cardccrd 9048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-rep 4965  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-pss 3786  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-tp 4374  df-op 4376  df-uni 4630  df-int 4669  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5221  df-eprel 5226  df-po 5234  df-so 5235  df-fr 5272  df-se 5273  df-we 5274  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-pred 5899  df-ord 5945  df-on 5946  df-suc 5948  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110  df-isom 6111  df-riota 6840  df-ov 6882  df-oprab 6883  df-mpt2 6884  df-1st 7402  df-2nd 7403  df-wrecs 7646  df-recs 7708  df-er 7983  df-map 8098  df-en 8197  df-dom 8198  df-sdom 8199  df-wdom 8707  df-card 9052  df-acn 9055
This theorem is referenced by:  wdomac  9638  ttac  38383  isnumbasgrplem2  38454
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