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Theorem cardiun 8846
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 7182 . . . . . 6 (𝐴𝑉 → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)} ∈ V)
2 vex 3234 . . . . . . . . 9 𝑦 ∈ V
3 eqeq1 2655 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝑧 = (card‘𝐵) ↔ 𝑦 = (card‘𝐵)))
43rexbidv 3081 . . . . . . . . 9 (𝑧 = 𝑦 → (∃𝑥𝐴 𝑧 = (card‘𝐵) ↔ ∃𝑥𝐴 𝑦 = (card‘𝐵)))
52, 4elab 3382 . . . . . . . 8 (𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)} ↔ ∃𝑥𝐴 𝑦 = (card‘𝐵))
6 cardidm 8823 . . . . . . . . . 10 (card‘(card‘𝐵)) = (card‘𝐵)
7 fveq2 6229 . . . . . . . . . 10 (𝑦 = (card‘𝐵) → (card‘𝑦) = (card‘(card‘𝐵)))
8 id 22 . . . . . . . . . 10 (𝑦 = (card‘𝐵) → 𝑦 = (card‘𝐵))
96, 7, 83eqtr4a 2711 . . . . . . . . 9 (𝑦 = (card‘𝐵) → (card‘𝑦) = 𝑦)
109rexlimivw 3058 . . . . . . . 8 (∃𝑥𝐴 𝑦 = (card‘𝐵) → (card‘𝑦) = 𝑦)
115, 10sylbi 207 . . . . . . 7 (𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)} → (card‘𝑦) = 𝑦)
1211rgen 2951 . . . . . 6 𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)} (card‘𝑦) = 𝑦
13 carduni 8845 . . . . . 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 6239 . . . . . . 7 (card‘𝐵) ∈ V
1615dfiun2 4586 . . . . . 6 𝑥𝐴 (card‘𝐵) = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)}
1716fveq2i 6232 . . . . 5 (card‘ 𝑥𝐴 (card‘𝐵)) = (card‘ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)})
1814, 17, 163eqtr4g 2710 . . . 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 4569 . . . . 5 (∀𝑥𝐴 (card‘𝐵) = 𝐵 𝑥𝐴 (card‘𝐵) = 𝑥𝐴 𝐵)
2120adantl 481 . . . 4 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝐵) = 𝐵) → 𝑥𝐴 (card‘𝐵) = 𝑥𝐴 𝐵)
2221fveq2d 6233 . . 3 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝐵) = 𝐵) → (card‘ 𝑥𝐴 (card‘𝐵)) = (card‘ 𝑥𝐴 𝐵))
2319, 22, 213eqtr3d 2693 . 2 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝐵) = 𝐵) → (card‘ 𝑥𝐴 𝐵) = 𝑥𝐴 𝐵)
2423ex 449 1 (𝐴𝑉 → (∀𝑥𝐴 (card‘𝐵) = 𝐵 → (card‘ 𝑥𝐴 𝐵) = 𝑥𝐴 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  {cab 2637  wral 2941  wrex 2942  Vcvv 3231   cuni 4468   ciun 4552  cfv 5926  cardccrd 8799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-card 8803
This theorem is referenced by:  alephcard  8931
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