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Theorem cardiun 9399
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 7648 . . . . . 6 (𝐴𝑉 → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)} ∈ V)
2 vex 3472 . . . . . . . . 9 𝑦 ∈ V
3 eqeq1 2826 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝑧 = (card‘𝐵) ↔ 𝑦 = (card‘𝐵)))
43rexbidv 3283 . . . . . . . . 9 (𝑧 = 𝑦 → (∃𝑥𝐴 𝑧 = (card‘𝐵) ↔ ∃𝑥𝐴 𝑦 = (card‘𝐵)))
52, 4elab 3642 . . . . . . . 8 (𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)} ↔ ∃𝑥𝐴 𝑦 = (card‘𝐵))
6 cardidm 9376 . . . . . . . . . 10 (card‘(card‘𝐵)) = (card‘𝐵)
7 fveq2 6652 . . . . . . . . . 10 (𝑦 = (card‘𝐵) → (card‘𝑦) = (card‘(card‘𝐵)))
8 id 22 . . . . . . . . . 10 (𝑦 = (card‘𝐵) → 𝑦 = (card‘𝐵))
96, 7, 83eqtr4a 2883 . . . . . . . . 9 (𝑦 = (card‘𝐵) → (card‘𝑦) = 𝑦)
109rexlimivw 3268 . . . . . . . 8 (∃𝑥𝐴 𝑦 = (card‘𝐵) → (card‘𝑦) = 𝑦)
115, 10sylbi 220 . . . . . . 7 (𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)} → (card‘𝑦) = 𝑦)
1211rgen 3140 . . . . . 6 𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)} (card‘𝑦) = 𝑦
13 carduni 9398 . . . . . 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 6665 . . . . . . 7 (card‘𝐵) ∈ V
1615dfiun2 4933 . . . . . 6 𝑥𝐴 (card‘𝐵) = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)}
1716fveq2i 6655 . . . . 5 (card‘ 𝑥𝐴 (card‘𝐵)) = (card‘ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)})
1814, 17, 163eqtr4g 2882 . . . 4 (𝐴𝑉 → (card‘ 𝑥𝐴 (card‘𝐵)) = 𝑥𝐴 (card‘𝐵))
1918adantr 484 . . 3 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝐵) = 𝐵) → (card‘ 𝑥𝐴 (card‘𝐵)) = 𝑥𝐴 (card‘𝐵))
20 iuneq2 4913 . . . . 5 (∀𝑥𝐴 (card‘𝐵) = 𝐵 𝑥𝐴 (card‘𝐵) = 𝑥𝐴 𝐵)
2120adantl 485 . . . 4 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝐵) = 𝐵) → 𝑥𝐴 (card‘𝐵) = 𝑥𝐴 𝐵)
2221fveq2d 6656 . . 3 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝐵) = 𝐵) → (card‘ 𝑥𝐴 (card‘𝐵)) = (card‘ 𝑥𝐴 𝐵))
2319, 22, 213eqtr3d 2865 . 2 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝐵) = 𝐵) → (card‘ 𝑥𝐴 𝐵) = 𝑥𝐴 𝐵)
2423ex 416 1 (𝐴𝑉 → (∀𝑥𝐴 (card‘𝐵) = 𝐵 → (card‘ 𝑥𝐴 𝐵) = 𝑥𝐴 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  {cab 2800  wral 3130  wrex 3131  Vcvv 3469   cuni 4813   ciun 4894  cfv 6334  cardccrd 9352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-ord 6172  df-on 6173  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-card 9356
This theorem is referenced by:  alephcard  9485
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