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Theorem harval2 9156
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 8756 . . . . . . 7 (𝐴 ∈ dom card → (har‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
21adantr 474 . . . . . 6 ((𝐴 ∈ dom card ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → (har‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
3 sdomel 8395 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦𝑥𝑦𝑥))
4 domsdomtr 8383 . . . . . . . . . . . 12 ((𝑦𝐴𝐴𝑥) → 𝑦𝑥)
53, 4impel 501 . . . . . . . . . . 11 (((𝑦 ∈ On ∧ 𝑥 ∈ On) ∧ (𝑦𝐴𝐴𝑥)) → 𝑦𝑥)
65an4s 650 . . . . . . . . . 10 (((𝑦 ∈ On ∧ 𝑦𝐴) ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → 𝑦𝑥)
76ancoms 452 . . . . . . . . 9 (((𝑥 ∈ On ∧ 𝐴𝑥) ∧ (𝑦 ∈ On ∧ 𝑦𝐴)) → 𝑦𝑥)
873impb 1104 . . . . . . . 8 (((𝑥 ∈ On ∧ 𝐴𝑥) ∧ 𝑦 ∈ On ∧ 𝑦𝐴) → 𝑦𝑥)
98rabssdv 3903 . . . . . . 7 ((𝑥 ∈ On ∧ 𝐴𝑥) → {𝑦 ∈ On ∣ 𝑦𝐴} ⊆ 𝑥)
109adantl 475 . . . . . 6 ((𝐴 ∈ dom card ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → {𝑦 ∈ On ∣ 𝑦𝐴} ⊆ 𝑥)
112, 10eqsstrd 3858 . . . . 5 ((𝐴 ∈ dom card ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → (har‘𝐴) ⊆ 𝑥)
1211expr 450 . . . 4 ((𝐴 ∈ dom card ∧ 𝑥 ∈ On) → (𝐴𝑥 → (har‘𝐴) ⊆ 𝑥))
1312ralrimiva 3148 . . 3 (𝐴 ∈ dom card → ∀𝑥 ∈ On (𝐴𝑥 → (har‘𝐴) ⊆ 𝑥))
14 ssintrab 4733 . . 3 ((har‘𝐴) ⊆ {𝑥 ∈ On ∣ 𝐴𝑥} ↔ ∀𝑥 ∈ On (𝐴𝑥 → (har‘𝐴) ⊆ 𝑥))
1513, 14sylibr 226 . 2 (𝐴 ∈ dom card → (har‘𝐴) ⊆ {𝑥 ∈ On ∣ 𝐴𝑥})
16 breq2 4890 . . . 4 (𝑥 = (har‘𝐴) → (𝐴𝑥𝐴 ≺ (har‘𝐴)))
17 harcl 8755 . . . . 5 (har‘𝐴) ∈ On
1817a1i 11 . . . 4 (𝐴 ∈ dom card → (har‘𝐴) ∈ On)
19 harsdom 9154 . . . 4 (𝐴 ∈ dom card → 𝐴 ≺ (har‘𝐴))
2016, 18, 19elrabd 3575 . . 3 (𝐴 ∈ dom card → (har‘𝐴) ∈ {𝑥 ∈ On ∣ 𝐴𝑥})
21 intss1 4725 . . 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 3838 1 (𝐴 ∈ dom card → (har‘𝐴) = {𝑥 ∈ On ∣ 𝐴𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  wral 3090  {crab 3094  wss 3792   cint 4710   class class class wbr 4886  dom cdm 5355  Oncon0 5976  cfv 6135  cdom 8239  csdm 8240  harchar 8750  cardccrd 9094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-wrecs 7689  df-recs 7751  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-oi 8704  df-har 8752  df-card 9098
This theorem is referenced by:  alephnbtwn  9227
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