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Theorem harval2 8775
 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 8419 . . . . . . 7 (𝐴 ∈ dom card → (har‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
21adantr 481 . . . . . 6 ((𝐴 ∈ dom card ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → (har‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
3 domsdomtr 8047 . . . . . . . . . . . . 13 ((𝑦𝐴𝐴𝑥) → 𝑦𝑥)
4 sdomel 8059 . . . . . . . . . . . . 13 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦𝑥𝑦𝑥))
53, 4syl5 34 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → ((𝑦𝐴𝐴𝑥) → 𝑦𝑥))
65imp 445 . . . . . . . . . . 11 (((𝑦 ∈ On ∧ 𝑥 ∈ On) ∧ (𝑦𝐴𝐴𝑥)) → 𝑦𝑥)
76an4s 868 . . . . . . . . . 10 (((𝑦 ∈ On ∧ 𝑦𝐴) ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → 𝑦𝑥)
87ancoms 469 . . . . . . . . 9 (((𝑥 ∈ On ∧ 𝐴𝑥) ∧ (𝑦 ∈ On ∧ 𝑦𝐴)) → 𝑦𝑥)
983impb 1257 . . . . . . . 8 (((𝑥 ∈ On ∧ 𝐴𝑥) ∧ 𝑦 ∈ On ∧ 𝑦𝐴) → 𝑦𝑥)
109rabssdv 3666 . . . . . . 7 ((𝑥 ∈ On ∧ 𝐴𝑥) → {𝑦 ∈ On ∣ 𝑦𝐴} ⊆ 𝑥)
1110adantl 482 . . . . . 6 ((𝐴 ∈ dom card ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → {𝑦 ∈ On ∣ 𝑦𝐴} ⊆ 𝑥)
122, 11eqsstrd 3623 . . . . 5 ((𝐴 ∈ dom card ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → (har‘𝐴) ⊆ 𝑥)
1312expr 642 . . . 4 ((𝐴 ∈ dom card ∧ 𝑥 ∈ On) → (𝐴𝑥 → (har‘𝐴) ⊆ 𝑥))
1413ralrimiva 2961 . . 3 (𝐴 ∈ dom card → ∀𝑥 ∈ On (𝐴𝑥 → (har‘𝐴) ⊆ 𝑥))
15 ssintrab 4470 . . 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 8418 . . . . 5 (har‘𝐴) ∈ On
1817a1i 11 . . . 4 (𝐴 ∈ dom card → (har‘𝐴) ∈ On)
19 harsdom 8773 . . . 4 (𝐴 ∈ dom card → 𝐴 ≺ (har‘𝐴))
20 breq2 4622 . . . . 5 (𝑥 = (har‘𝐴) → (𝐴𝑥𝐴 ≺ (har‘𝐴)))
2120elrab 3350 . . . 4 ((har‘𝐴) ∈ {𝑥 ∈ On ∣ 𝐴𝑥} ↔ ((har‘𝐴) ∈ On ∧ 𝐴 ≺ (har‘𝐴)))
2218, 19, 21sylanbrc 697 . . 3 (𝐴 ∈ dom card → (har‘𝐴) ∈ {𝑥 ∈ On ∣ 𝐴𝑥})
23 intss1 4462 . . 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 3604 1 (𝐴 ∈ dom card → (har‘𝐴) = {𝑥 ∈ On ∣ 𝐴𝑥})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∀wral 2907  {crab 2911   ⊆ wss 3559  ∩ cint 4445   class class class wbr 4618  dom cdm 5079  Oncon0 5687  ‘cfv 5852   ≼ cdom 7905   ≺ csdm 7906  harchar 8413  cardccrd 8713 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-wrecs 7359  df-recs 7420  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-oi 8367  df-har 8415  df-card 8717 This theorem is referenced by:  alephnbtwn  8846
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