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Theorem harval3 43556
Description: (har‘𝐴) is the least cardinal that is greater than 𝐴. (Contributed by RP, 4-Nov-2023.)
Assertion
Ref Expression
harval3 (𝐴 ∈ dom card → (har‘𝐴) = {𝑥 ∈ ran card ∣ 𝐴𝑥})
Distinct variable group:   𝑥,𝐴

Proof of Theorem harval3
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 harval2 10038 . 2 (𝐴 ∈ dom card → (har‘𝐴) = {𝑦 ∈ On ∣ 𝐴𝑦})
2 vex 3483 . . . . . 6 𝑥 ∈ V
32a1i 11 . . . . 5 (𝐴 ∈ dom card → 𝑥 ∈ V)
4 elrncard 43555 . . . . . . . . 9 (𝑥 ∈ ran card ↔ (𝑥 ∈ On ∧ ∀𝑦𝑥 ¬ 𝑦𝑥))
54simplbi 497 . . . . . . . 8 (𝑥 ∈ ran card → 𝑥 ∈ On)
65anim1i 615 . . . . . . 7 ((𝑥 ∈ ran card ∧ 𝐴𝑥) → (𝑥 ∈ On ∧ 𝐴𝑥))
7 eleq1 2828 . . . . . . . 8 (𝑦 = 𝑥 → (𝑦 ∈ On ↔ 𝑥 ∈ On))
8 breq2 5146 . . . . . . . 8 (𝑦 = 𝑥 → (𝐴𝑦𝐴𝑥))
97, 8anbi12d 632 . . . . . . 7 (𝑦 = 𝑥 → ((𝑦 ∈ On ∧ 𝐴𝑦) ↔ (𝑥 ∈ On ∧ 𝐴𝑥)))
106, 9imbitrrid 246 . . . . . 6 (𝑦 = 𝑥 → ((𝑥 ∈ ran card ∧ 𝐴𝑥) → (𝑦 ∈ On ∧ 𝐴𝑦)))
1110adantl 481 . . . . 5 ((𝐴 ∈ dom card ∧ 𝑦 = 𝑥) → ((𝑥 ∈ ran card ∧ 𝐴𝑥) → (𝑦 ∈ On ∧ 𝐴𝑦)))
12 ssidd 4006 . . . . 5 (𝐴 ∈ dom card → 𝑥𝑥)
133, 11, 12intabssd 43537 . . . 4 (𝐴 ∈ dom card → {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴𝑦)} ⊆ {𝑥 ∣ (𝑥 ∈ ran card ∧ 𝐴𝑥)})
14 vex 3483 . . . . . . 7 𝑦 ∈ V
1514inex1 5316 . . . . . 6 (𝑦 ∩ (card‘𝑦)) ∈ V
1615a1i 11 . . . . 5 (𝐴 ∈ dom card → (𝑦 ∩ (card‘𝑦)) ∈ V)
17 oncardid 9997 . . . . . . . . . . . 12 (𝑦 ∈ On → (card‘𝑦) ≈ 𝑦)
1817ensymd 9046 . . . . . . . . . . 11 (𝑦 ∈ On → 𝑦 ≈ (card‘𝑦))
19 sdomentr 9152 . . . . . . . . . . . 12 ((𝐴𝑦𝑦 ≈ (card‘𝑦)) → 𝐴 ≺ (card‘𝑦))
2019a1i 11 . . . . . . . . . . 11 (𝑦 ∈ On → ((𝐴𝑦𝑦 ≈ (card‘𝑦)) → 𝐴 ≺ (card‘𝑦)))
2118, 20mpan2d 694 . . . . . . . . . 10 (𝑦 ∈ On → (𝐴𝑦𝐴 ≺ (card‘𝑦)))
22 df-card 9980 . . . . . . . . . . . 12 card = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
2322funmpt2 6604 . . . . . . . . . . 11 Fun card
24 onenon 9990 . . . . . . . . . . 11 (𝑦 ∈ On → 𝑦 ∈ dom card)
25 fvelrn 7095 . . . . . . . . . . 11 ((Fun card ∧ 𝑦 ∈ dom card) → (card‘𝑦) ∈ ran card)
2623, 24, 25sylancr 587 . . . . . . . . . 10 (𝑦 ∈ On → (card‘𝑦) ∈ ran card)
2721, 26jctild 525 . . . . . . . . 9 (𝑦 ∈ On → (𝐴𝑦 → ((card‘𝑦) ∈ ran card ∧ 𝐴 ≺ (card‘𝑦))))
2827adantl 481 . . . . . . . 8 ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → (𝐴𝑦 → ((card‘𝑦) ∈ ran card ∧ 𝐴 ≺ (card‘𝑦))))
29 simpl 482 . . . . . . . . . 10 ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → 𝑥 = (𝑦 ∩ (card‘𝑦)))
30 cardonle 9998 . . . . . . . . . . . 12 (𝑦 ∈ On → (card‘𝑦) ⊆ 𝑦)
3130adantl 481 . . . . . . . . . . 11 ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → (card‘𝑦) ⊆ 𝑦)
32 sseqin2 4222 . . . . . . . . . . 11 ((card‘𝑦) ⊆ 𝑦 ↔ (𝑦 ∩ (card‘𝑦)) = (card‘𝑦))
3331, 32sylib 218 . . . . . . . . . 10 ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → (𝑦 ∩ (card‘𝑦)) = (card‘𝑦))
3429, 33eqtrd 2776 . . . . . . . . 9 ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → 𝑥 = (card‘𝑦))
35 eleq1 2828 . . . . . . . . . 10 (𝑥 = (card‘𝑦) → (𝑥 ∈ ran card ↔ (card‘𝑦) ∈ ran card))
36 breq2 5146 . . . . . . . . . 10 (𝑥 = (card‘𝑦) → (𝐴𝑥𝐴 ≺ (card‘𝑦)))
3735, 36anbi12d 632 . . . . . . . . 9 (𝑥 = (card‘𝑦) → ((𝑥 ∈ ran card ∧ 𝐴𝑥) ↔ ((card‘𝑦) ∈ ran card ∧ 𝐴 ≺ (card‘𝑦))))
3834, 37syl 17 . . . . . . . 8 ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → ((𝑥 ∈ ran card ∧ 𝐴𝑥) ↔ ((card‘𝑦) ∈ ran card ∧ 𝐴 ≺ (card‘𝑦))))
3928, 38sylibrd 259 . . . . . . 7 ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → (𝐴𝑦 → (𝑥 ∈ ran card ∧ 𝐴𝑥)))
4039expimpd 453 . . . . . 6 (𝑥 = (𝑦 ∩ (card‘𝑦)) → ((𝑦 ∈ On ∧ 𝐴𝑦) → (𝑥 ∈ ran card ∧ 𝐴𝑥)))
4140adantl 481 . . . . 5 ((𝐴 ∈ dom card ∧ 𝑥 = (𝑦 ∩ (card‘𝑦))) → ((𝑦 ∈ On ∧ 𝐴𝑦) → (𝑥 ∈ ran card ∧ 𝐴𝑥)))
42 inss1 4236 . . . . . 6 (𝑦 ∩ (card‘𝑦)) ⊆ 𝑦
4342a1i 11 . . . . 5 (𝐴 ∈ dom card → (𝑦 ∩ (card‘𝑦)) ⊆ 𝑦)
4416, 41, 43intabssd 43537 . . . 4 (𝐴 ∈ dom card → {𝑥 ∣ (𝑥 ∈ ran card ∧ 𝐴𝑥)} ⊆ {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴𝑦)})
4513, 44eqssd 4000 . . 3 (𝐴 ∈ dom card → {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴𝑦)} = {𝑥 ∣ (𝑥 ∈ ran card ∧ 𝐴𝑥)})
46 df-rab 3436 . . . 4 {𝑦 ∈ On ∣ 𝐴𝑦} = {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴𝑦)}
4746inteqi 4949 . . 3 {𝑦 ∈ On ∣ 𝐴𝑦} = {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴𝑦)}
48 df-rab 3436 . . . 4 {𝑥 ∈ ran card ∣ 𝐴𝑥} = {𝑥 ∣ (𝑥 ∈ ran card ∧ 𝐴𝑥)}
4948inteqi 4949 . . 3 {𝑥 ∈ ran card ∣ 𝐴𝑥} = {𝑥 ∣ (𝑥 ∈ ran card ∧ 𝐴𝑥)}
5045, 47, 493eqtr4g 2801 . 2 (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝐴𝑦} = {𝑥 ∈ ran card ∣ 𝐴𝑥})
511, 50eqtrd 2776 1 (𝐴 ∈ dom card → (har‘𝐴) = {𝑥 ∈ ran card ∣ 𝐴𝑥})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  {cab 2713  wral 3060  {crab 3435  Vcvv 3479  cin 3949  wss 3950   cint 4945   class class class wbr 5142  dom cdm 5684  ran crn 5685  Oncon0 6383  Fun wfun 6554  cfv 6560  cen 8983  csdm 8985  harchar 9597  cardccrd 9976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-oi 9551  df-har 9598  df-card 9980
This theorem is referenced by:  harval3on  43557
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