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Theorem harval2 9425
 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 9025 . . . . . . 7 (𝐴 ∈ dom card → (har‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
21adantr 483 . . . . . 6 ((𝐴 ∈ dom card ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → (har‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
3 sdomel 8663 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦𝑥𝑦𝑥))
4 domsdomtr 8651 . . . . . . . . . . . 12 ((𝑦𝐴𝐴𝑥) → 𝑦𝑥)
53, 4impel 508 . . . . . . . . . . 11 (((𝑦 ∈ On ∧ 𝑥 ∈ On) ∧ (𝑦𝐴𝐴𝑥)) → 𝑦𝑥)
65an4s 658 . . . . . . . . . 10 (((𝑦 ∈ On ∧ 𝑦𝐴) ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → 𝑦𝑥)
76ancoms 461 . . . . . . . . 9 (((𝑥 ∈ On ∧ 𝐴𝑥) ∧ (𝑦 ∈ On ∧ 𝑦𝐴)) → 𝑦𝑥)
873impb 1111 . . . . . . . 8 (((𝑥 ∈ On ∧ 𝐴𝑥) ∧ 𝑦 ∈ On ∧ 𝑦𝐴) → 𝑦𝑥)
98rabssdv 4050 . . . . . . 7 ((𝑥 ∈ On ∧ 𝐴𝑥) → {𝑦 ∈ On ∣ 𝑦𝐴} ⊆ 𝑥)
109adantl 484 . . . . . 6 ((𝐴 ∈ dom card ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → {𝑦 ∈ On ∣ 𝑦𝐴} ⊆ 𝑥)
112, 10eqsstrd 4004 . . . . 5 ((𝐴 ∈ dom card ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → (har‘𝐴) ⊆ 𝑥)
1211expr 459 . . . 4 ((𝐴 ∈ dom card ∧ 𝑥 ∈ On) → (𝐴𝑥 → (har‘𝐴) ⊆ 𝑥))
1312ralrimiva 3182 . . 3 (𝐴 ∈ dom card → ∀𝑥 ∈ On (𝐴𝑥 → (har‘𝐴) ⊆ 𝑥))
14 ssintrab 4898 . . 3 ((har‘𝐴) ⊆ {𝑥 ∈ On ∣ 𝐴𝑥} ↔ ∀𝑥 ∈ On (𝐴𝑥 → (har‘𝐴) ⊆ 𝑥))
1513, 14sylibr 236 . 2 (𝐴 ∈ dom card → (har‘𝐴) ⊆ {𝑥 ∈ On ∣ 𝐴𝑥})
16 breq2 5069 . . . 4 (𝑥 = (har‘𝐴) → (𝐴𝑥𝐴 ≺ (har‘𝐴)))
17 harcl 9024 . . . . 5 (har‘𝐴) ∈ On
1817a1i 11 . . . 4 (𝐴 ∈ dom card → (har‘𝐴) ∈ On)
19 harsdom 9423 . . . 4 (𝐴 ∈ dom card → 𝐴 ≺ (har‘𝐴))
2016, 18, 19elrabd 3681 . . 3 (𝐴 ∈ dom card → (har‘𝐴) ∈ {𝑥 ∈ On ∣ 𝐴𝑥})
21 intss1 4890 . . 3 ((har‘𝐴) ∈ {𝑥 ∈ On ∣ 𝐴𝑥} → {𝑥 ∈ On ∣ 𝐴𝑥} ⊆ (har‘𝐴))
2220, 21syl 17 . 2 (𝐴 ∈ dom card → {𝑥 ∈ On ∣ 𝐴𝑥} ⊆ (har‘𝐴))
2315, 22eqssd 3983 1 (𝐴 ∈ dom card → (har‘𝐴) = {𝑥 ∈ On ∣ 𝐴𝑥})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  {crab 3142   ⊆ wss 3935  ∩ cint 4875   class class class wbr 5065  dom cdm 5554  Oncon0 6190  ‘cfv 6354   ≼ cdom 8506   ≺ csdm 8507  harchar 9019  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-lim 6195  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-wrecs 7946  df-recs 8007  df-er 8288  df-en 8509  df-dom 8510  df-sdom 8511  df-oi 8973  df-har 9021  df-card 9367 This theorem is referenced by:  alephnbtwn  9496  harsucnn  39903  harval3  39904  aleph1min  39916
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