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Theorem cardiun 10023
Description: The indexed union of a set of cardinals is a cardinal. (Contributed by NM, 3-Nov-2003.)
Assertion
Ref Expression
cardiun (𝐴𝑉 → (∀𝑥𝐴 (card‘𝐵) = 𝐵 → (card‘ 𝑥𝐴 𝐵) = 𝑥𝐴 𝐵))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem cardiun
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 abrexexg 7986 . . . . . 6 (𝐴𝑉 → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)} ∈ V)
2 vex 3483 . . . . . . . . 9 𝑦 ∈ V
3 eqeq1 2740 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝑧 = (card‘𝐵) ↔ 𝑦 = (card‘𝐵)))
43rexbidv 3178 . . . . . . . . 9 (𝑧 = 𝑦 → (∃𝑥𝐴 𝑧 = (card‘𝐵) ↔ ∃𝑥𝐴 𝑦 = (card‘𝐵)))
52, 4elab 3678 . . . . . . . 8 (𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)} ↔ ∃𝑥𝐴 𝑦 = (card‘𝐵))
6 cardidm 10000 . . . . . . . . . 10 (card‘(card‘𝐵)) = (card‘𝐵)
7 fveq2 6905 . . . . . . . . . 10 (𝑦 = (card‘𝐵) → (card‘𝑦) = (card‘(card‘𝐵)))
8 id 22 . . . . . . . . . 10 (𝑦 = (card‘𝐵) → 𝑦 = (card‘𝐵))
96, 7, 83eqtr4a 2802 . . . . . . . . 9 (𝑦 = (card‘𝐵) → (card‘𝑦) = 𝑦)
109rexlimivw 3150 . . . . . . . 8 (∃𝑥𝐴 𝑦 = (card‘𝐵) → (card‘𝑦) = 𝑦)
115, 10sylbi 217 . . . . . . 7 (𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)} → (card‘𝑦) = 𝑦)
1211rgen 3062 . . . . . 6 𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)} (card‘𝑦) = 𝑦
13 carduni 10022 . . . . . 6 ({𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)} ∈ V → (∀𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)} (card‘𝑦) = 𝑦 → (card‘ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)}) = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)}))
141, 12, 13mpisyl 21 . . . . 5 (𝐴𝑉 → (card‘ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)}) = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)})
15 fvex 6918 . . . . . . 7 (card‘𝐵) ∈ V
1615dfiun2 5032 . . . . . 6 𝑥𝐴 (card‘𝐵) = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)}
1716fveq2i 6908 . . . . 5 (card‘ 𝑥𝐴 (card‘𝐵)) = (card‘ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)})
1814, 17, 163eqtr4g 2801 . . . 4 (𝐴𝑉 → (card‘ 𝑥𝐴 (card‘𝐵)) = 𝑥𝐴 (card‘𝐵))
1918adantr 480 . . 3 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝐵) = 𝐵) → (card‘ 𝑥𝐴 (card‘𝐵)) = 𝑥𝐴 (card‘𝐵))
20 iuneq2 5010 . . . . 5 (∀𝑥𝐴 (card‘𝐵) = 𝐵 𝑥𝐴 (card‘𝐵) = 𝑥𝐴 𝐵)
2120adantl 481 . . . 4 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝐵) = 𝐵) → 𝑥𝐴 (card‘𝐵) = 𝑥𝐴 𝐵)
2221fveq2d 6909 . . 3 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝐵) = 𝐵) → (card‘ 𝑥𝐴 (card‘𝐵)) = (card‘ 𝑥𝐴 𝐵))
2319, 22, 213eqtr3d 2784 . 2 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝐵) = 𝐵) → (card‘ 𝑥𝐴 𝐵) = 𝑥𝐴 𝐵)
2423ex 412 1 (𝐴𝑉 → (∀𝑥𝐴 (card‘𝐵) = 𝐵 → (card‘ 𝑥𝐴 𝐵) = 𝑥𝐴 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  {cab 2713  wral 3060  wrex 3069  Vcvv 3479   cuni 4906   ciun 4990  cfv 6560  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-rab 3436  df-v 3481  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-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-ord 6386  df-on 6387  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-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-card 9980
This theorem is referenced by:  alephcard  10111
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