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Theorem wdomnumr 8872
 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 8457 . . 3 (𝐵 ∈ dom card → (𝐴* 𝐵 ↔ (𝐴 = ∅ ∨ ∃𝑥 𝑥:𝐵onto𝐴)))
2 0domg 8072 . . . . 5 (𝐵 ∈ dom card → ∅ ≼ 𝐵)
3 breq1 4647 . . . . 5 (𝐴 = ∅ → (𝐴𝐵 ↔ ∅ ≼ 𝐵))
42, 3syl5ibrcom 237 . . . 4 (𝐵 ∈ dom card → (𝐴 = ∅ → 𝐴𝐵))
5 fodomnum 8865 . . . . 5 (𝐵 ∈ dom card → (𝑥:𝐵onto𝐴𝐴𝐵))
65exlimdv 1859 . . . 4 (𝐵 ∈ dom card → (∃𝑥 𝑥:𝐵onto𝐴𝐴𝐵))
74, 6jaod 395 . . 3 (𝐵 ∈ dom card → ((𝐴 = ∅ ∨ ∃𝑥 𝑥:𝐵onto𝐴) → 𝐴𝐵))
81, 7sylbid 230 . 2 (𝐵 ∈ dom card → (𝐴* 𝐵𝐴𝐵))
9 domwdom 8464 . 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 383   = wceq 1481  ∃wex 1702   ∈ wcel 1988  ∅c0 3907   class class class wbr 4644  dom cdm 5104  –onto→wfo 5874   ≼ cdom 7938   ≼* cwdom 8447  cardccrd 8746 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-wdom 8449  df-card 8750  df-acn 8753 This theorem is referenced by:  wdomac  9334  ttac  37422  isnumbasgrplem2  37493
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