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Theorem cardiun 10020
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 7984 . . . . . 6 (𝐴𝑉 → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)} ∈ V)
2 vex 3482 . . . . . . . . 9 𝑦 ∈ V
3 eqeq1 2739 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝑧 = (card‘𝐵) ↔ 𝑦 = (card‘𝐵)))
43rexbidv 3177 . . . . . . . . 9 (𝑧 = 𝑦 → (∃𝑥𝐴 𝑧 = (card‘𝐵) ↔ ∃𝑥𝐴 𝑦 = (card‘𝐵)))
52, 4elab 3681 . . . . . . . 8 (𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)} ↔ ∃𝑥𝐴 𝑦 = (card‘𝐵))
6 cardidm 9997 . . . . . . . . . 10 (card‘(card‘𝐵)) = (card‘𝐵)
7 fveq2 6907 . . . . . . . . . 10 (𝑦 = (card‘𝐵) → (card‘𝑦) = (card‘(card‘𝐵)))
8 id 22 . . . . . . . . . 10 (𝑦 = (card‘𝐵) → 𝑦 = (card‘𝐵))
96, 7, 83eqtr4a 2801 . . . . . . . . 9 (𝑦 = (card‘𝐵) → (card‘𝑦) = 𝑦)
109rexlimivw 3149 . . . . . . . 8 (∃𝑥𝐴 𝑦 = (card‘𝐵) → (card‘𝑦) = 𝑦)
115, 10sylbi 217 . . . . . . 7 (𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)} → (card‘𝑦) = 𝑦)
1211rgen 3061 . . . . . 6 𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)} (card‘𝑦) = 𝑦
13 carduni 10019 . . . . . 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 6920 . . . . . . 7 (card‘𝐵) ∈ V
1615dfiun2 5038 . . . . . 6 𝑥𝐴 (card‘𝐵) = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)}
1716fveq2i 6910 . . . . 5 (card‘ 𝑥𝐴 (card‘𝐵)) = (card‘ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)})
1814, 17, 163eqtr4g 2800 . . . 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 5016 . . . . 5 (∀𝑥𝐴 (card‘𝐵) = 𝐵 𝑥𝐴 (card‘𝐵) = 𝑥𝐴 𝐵)
2120adantl 481 . . . 4 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝐵) = 𝐵) → 𝑥𝐴 (card‘𝐵) = 𝑥𝐴 𝐵)
2221fveq2d 6911 . . 3 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝐵) = 𝐵) → (card‘ 𝑥𝐴 (card‘𝐵)) = (card‘ 𝑥𝐴 𝐵))
2319, 22, 213eqtr3d 2783 . 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 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  Vcvv 3478   cuni 4912   ciun 4996  cfv 6563  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-rab 3434  df-v 3480  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-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-ord 6389  df-on 6390  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-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-card 9977
This theorem is referenced by:  alephcard  10108
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