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Theorem harval2 8861
Description: An alternate expression for the Hartogs number of a well-orderable set. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
harval2 (𝐴 ∈ dom card → (har‘𝐴) = {𝑥 ∈ On ∣ 𝐴𝑥})
Distinct variable group:   𝑥,𝐴

Proof of Theorem harval2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 harval 8508 . . . . . . 7 (𝐴 ∈ dom card → (har‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
21adantr 480 . . . . . 6 ((𝐴 ∈ dom card ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → (har‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
3 domsdomtr 8136 . . . . . . . . . . . . 13 ((𝑦𝐴𝐴𝑥) → 𝑦𝑥)
4 sdomel 8148 . . . . . . . . . . . . 13 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦𝑥𝑦𝑥))
53, 4syl5 34 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → ((𝑦𝐴𝐴𝑥) → 𝑦𝑥))
65imp 444 . . . . . . . . . . 11 (((𝑦 ∈ On ∧ 𝑥 ∈ On) ∧ (𝑦𝐴𝐴𝑥)) → 𝑦𝑥)
76an4s 886 . . . . . . . . . 10 (((𝑦 ∈ On ∧ 𝑦𝐴) ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → 𝑦𝑥)
87ancoms 468 . . . . . . . . 9 (((𝑥 ∈ On ∧ 𝐴𝑥) ∧ (𝑦 ∈ On ∧ 𝑦𝐴)) → 𝑦𝑥)
983impb 1279 . . . . . . . 8 (((𝑥 ∈ On ∧ 𝐴𝑥) ∧ 𝑦 ∈ On ∧ 𝑦𝐴) → 𝑦𝑥)
109rabssdv 3715 . . . . . . 7 ((𝑥 ∈ On ∧ 𝐴𝑥) → {𝑦 ∈ On ∣ 𝑦𝐴} ⊆ 𝑥)
1110adantl 481 . . . . . 6 ((𝐴 ∈ dom card ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → {𝑦 ∈ On ∣ 𝑦𝐴} ⊆ 𝑥)
122, 11eqsstrd 3672 . . . . 5 ((𝐴 ∈ dom card ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → (har‘𝐴) ⊆ 𝑥)
1312expr 642 . . . 4 ((𝐴 ∈ dom card ∧ 𝑥 ∈ On) → (𝐴𝑥 → (har‘𝐴) ⊆ 𝑥))
1413ralrimiva 2995 . . 3 (𝐴 ∈ dom card → ∀𝑥 ∈ On (𝐴𝑥 → (har‘𝐴) ⊆ 𝑥))
15 ssintrab 4532 . . 3 ((har‘𝐴) ⊆ {𝑥 ∈ On ∣ 𝐴𝑥} ↔ ∀𝑥 ∈ On (𝐴𝑥 → (har‘𝐴) ⊆ 𝑥))
1614, 15sylibr 224 . 2 (𝐴 ∈ dom card → (har‘𝐴) ⊆ {𝑥 ∈ On ∣ 𝐴𝑥})
17 harcl 8507 . . . . 5 (har‘𝐴) ∈ On
1817a1i 11 . . . 4 (𝐴 ∈ dom card → (har‘𝐴) ∈ On)
19 harsdom 8859 . . . 4 (𝐴 ∈ dom card → 𝐴 ≺ (har‘𝐴))
20 breq2 4689 . . . . 5 (𝑥 = (har‘𝐴) → (𝐴𝑥𝐴 ≺ (har‘𝐴)))
2120elrab 3396 . . . 4 ((har‘𝐴) ∈ {𝑥 ∈ On ∣ 𝐴𝑥} ↔ ((har‘𝐴) ∈ On ∧ 𝐴 ≺ (har‘𝐴)))
2218, 19, 21sylanbrc 699 . . 3 (𝐴 ∈ dom card → (har‘𝐴) ∈ {𝑥 ∈ On ∣ 𝐴𝑥})
23 intss1 4524 . . 3 ((har‘𝐴) ∈ {𝑥 ∈ On ∣ 𝐴𝑥} → {𝑥 ∈ On ∣ 𝐴𝑥} ⊆ (har‘𝐴))
2422, 23syl 17 . 2 (𝐴 ∈ dom card → {𝑥 ∈ On ∣ 𝐴𝑥} ⊆ (har‘𝐴))
2516, 24eqssd 3653 1 (𝐴 ∈ dom card → (har‘𝐴) = {𝑥 ∈ On ∣ 𝐴𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  {crab 2945  wss 3607   cint 4507   class class class wbr 4685  dom cdm 5143  Oncon0 5761  cfv 5926  cdom 7995  csdm 7996  harchar 8502  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-lim 5766  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-wrecs 7452  df-recs 7513  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-oi 8456  df-har 8504  df-card 8803
This theorem is referenced by:  alephnbtwn  8932
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