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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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