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Theorem harval3 43528
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 10035 . 2 (𝐴 ∈ dom card → (har‘𝐴) = {𝑦 ∈ On ∣ 𝐴𝑦})
2 vex 3482 . . . . . 6 𝑥 ∈ V
32a1i 11 . . . . 5 (𝐴 ∈ dom card → 𝑥 ∈ V)
4 elrncard 43527 . . . . . . . . 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 2827 . . . . . . . 8 (𝑦 = 𝑥 → (𝑦 ∈ On ↔ 𝑥 ∈ On))
8 breq2 5152 . . . . . . . 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 4019 . . . . 5 (𝐴 ∈ dom card → 𝑥𝑥)
133, 11, 12intabssd 43509 . . . 4 (𝐴 ∈ dom card → {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴𝑦)} ⊆ {𝑥 ∣ (𝑥 ∈ ran card ∧ 𝐴𝑥)})
14 vex 3482 . . . . . . 7 𝑦 ∈ V
1514inex1 5323 . . . . . 6 (𝑦 ∩ (card‘𝑦)) ∈ V
1615a1i 11 . . . . 5 (𝐴 ∈ dom card → (𝑦 ∩ (card‘𝑦)) ∈ V)
17 oncardid 9994 . . . . . . . . . . . 12 (𝑦 ∈ On → (card‘𝑦) ≈ 𝑦)
1817ensymd 9044 . . . . . . . . . . 11 (𝑦 ∈ On → 𝑦 ≈ (card‘𝑦))
19 sdomentr 9150 . . . . . . . . . . . 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 9977 . . . . . . . . . . . 12 card = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
2322funmpt2 6607 . . . . . . . . . . 11 Fun card
24 onenon 9987 . . . . . . . . . . 11 (𝑦 ∈ On → 𝑦 ∈ dom card)
25 fvelrn 7096 . . . . . . . . . . 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 9995 . . . . . . . . . . . 12 (𝑦 ∈ On → (card‘𝑦) ⊆ 𝑦)
3130adantl 481 . . . . . . . . . . 11 ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → (card‘𝑦) ⊆ 𝑦)
32 sseqin2 4231 . . . . . . . . . . 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 2775 . . . . . . . . 9 ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → 𝑥 = (card‘𝑦))
35 eleq1 2827 . . . . . . . . . 10 (𝑥 = (card‘𝑦) → (𝑥 ∈ ran card ↔ (card‘𝑦) ∈ ran card))
36 breq2 5152 . . . . . . . . . 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 4245 . . . . . 6 (𝑦 ∩ (card‘𝑦)) ⊆ 𝑦
4342a1i 11 . . . . 5 (𝐴 ∈ dom card → (𝑦 ∩ (card‘𝑦)) ⊆ 𝑦)
4416, 41, 43intabssd 43509 . . . 4 (𝐴 ∈ dom card → {𝑥 ∣ (𝑥 ∈ ran card ∧ 𝐴𝑥)} ⊆ {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴𝑦)})
4513, 44eqssd 4013 . . 3 (𝐴 ∈ dom card → {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴𝑦)} = {𝑥 ∣ (𝑥 ∈ ran card ∧ 𝐴𝑥)})
46 df-rab 3434 . . . 4 {𝑦 ∈ On ∣ 𝐴𝑦} = {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴𝑦)}
4746inteqi 4955 . . 3 {𝑦 ∈ On ∣ 𝐴𝑦} = {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴𝑦)}
48 df-rab 3434 . . . 4 {𝑥 ∈ ran card ∣ 𝐴𝑥} = {𝑥 ∣ (𝑥 ∈ ran card ∧ 𝐴𝑥)}
4948inteqi 4955 . . 3 {𝑥 ∈ ran card ∣ 𝐴𝑥} = {𝑥 ∣ (𝑥 ∈ ran card ∧ 𝐴𝑥)}
5045, 47, 493eqtr4g 2800 . 2 (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝐴𝑦} = {𝑥 ∈ ran card ∣ 𝐴𝑥})
511, 50eqtrd 2775 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 1537  wcel 2106  {cab 2712  wral 3059  {crab 3433  Vcvv 3478  cin 3962  wss 3963   cint 4951   class class class wbr 5148  dom cdm 5689  ran crn 5690  Oncon0 6386  Fun wfun 6557  cfv 6563  cen 8981  csdm 8983  harchar 9594  cardccrd 9973
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-oi 9548  df-har 9595  df-card 9977
This theorem is referenced by:  harval3on  43529
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