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Theorem isnumi 6319
Description: A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
isnumi ((𝐴 ∈ On ∧ 𝐴𝐵) → 𝐵 ∈ dom card)

Proof of Theorem isnumi
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 3764 . . . . 5 (𝑦 = 𝐴 → (𝑦𝐵𝐴𝐵))
21rspcev 2653 . . . 4 ((𝐴 ∈ On ∧ 𝐴𝐵) → ∃𝑦 ∈ On 𝑦𝐵)
3 intexrabim 3904 . . . 4 (∃𝑦 ∈ On 𝑦𝐵 {𝑦 ∈ On ∣ 𝑦𝐵} ∈ V)
42, 3syl 14 . . 3 ((𝐴 ∈ On ∧ 𝐴𝐵) → {𝑦 ∈ On ∣ 𝑦𝐵} ∈ V)
5 encv 6190 . . . . . 6 (𝐴𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
65simprd 107 . . . . 5 (𝐴𝐵𝐵 ∈ V)
7 breq2 3765 . . . . . . . . 9 (𝑥 = 𝐵 → (𝑦𝑥𝑦𝐵))
87rabbidv 2546 . . . . . . . 8 (𝑥 = 𝐵 → {𝑦 ∈ On ∣ 𝑦𝑥} = {𝑦 ∈ On ∣ 𝑦𝐵})
98inteqd 3617 . . . . . . 7 (𝑥 = 𝐵 {𝑦 ∈ On ∣ 𝑦𝑥} = {𝑦 ∈ On ∣ 𝑦𝐵})
109eleq1d 2106 . . . . . 6 (𝑥 = 𝐵 → ( {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V ↔ {𝑦 ∈ On ∣ 𝑦𝐵} ∈ V))
1110elrab3 2696 . . . . 5 (𝐵 ∈ V → (𝐵 ∈ {𝑥 ∈ V ∣ {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V} ↔ {𝑦 ∈ On ∣ 𝑦𝐵} ∈ V))
126, 11syl 14 . . . 4 (𝐴𝐵 → (𝐵 ∈ {𝑥 ∈ V ∣ {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V} ↔ {𝑦 ∈ On ∣ 𝑦𝐵} ∈ V))
1312adantl 262 . . 3 ((𝐴 ∈ On ∧ 𝐴𝐵) → (𝐵 ∈ {𝑥 ∈ V ∣ {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V} ↔ {𝑦 ∈ On ∣ 𝑦𝐵} ∈ V))
144, 13mpbird 156 . 2 ((𝐴 ∈ On ∧ 𝐴𝐵) → 𝐵 ∈ {𝑥 ∈ V ∣ {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V})
15 df-card 6317 . . 3 card = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
1615dmmpt 4779 . 2 dom card = {𝑥 ∈ V ∣ {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V}
1714, 16syl6eleqr 2131 1 ((𝐴 ∈ On ∧ 𝐴𝐵) → 𝐵 ∈ dom card)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wcel 1393  wrex 2304  {crab 2307  Vcvv 2554   cint 3612   class class class wbr 3761  Oncon0 4072  dom cdm 4308  cen 6182  cardccrd 6316
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872  ax-pow 3924  ax-pr 3941
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-rab 2312  df-v 2556  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-int 3613  df-br 3762  df-opab 3816  df-mpt 3817  df-xp 4314  df-rel 4315  df-cnv 4316  df-dm 4318  df-rn 4319  df-res 4320  df-ima 4321  df-en 6185  df-card 6317
This theorem is referenced by:  finnum  6320  onenon  6321
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