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Theorem cardiun 8753
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 7091 . . . . . 6 (𝐴𝑉 → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)} ∈ V)
2 vex 3194 . . . . . . . . 9 𝑦 ∈ V
3 eqeq1 2630 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝑧 = (card‘𝐵) ↔ 𝑦 = (card‘𝐵)))
43rexbidv 3050 . . . . . . . . 9 (𝑧 = 𝑦 → (∃𝑥𝐴 𝑧 = (card‘𝐵) ↔ ∃𝑥𝐴 𝑦 = (card‘𝐵)))
52, 4elab 3338 . . . . . . . 8 (𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)} ↔ ∃𝑥𝐴 𝑦 = (card‘𝐵))
6 cardidm 8730 . . . . . . . . . 10 (card‘(card‘𝐵)) = (card‘𝐵)
7 fveq2 6150 . . . . . . . . . 10 (𝑦 = (card‘𝐵) → (card‘𝑦) = (card‘(card‘𝐵)))
8 id 22 . . . . . . . . . 10 (𝑦 = (card‘𝐵) → 𝑦 = (card‘𝐵))
96, 7, 83eqtr4a 2686 . . . . . . . . 9 (𝑦 = (card‘𝐵) → (card‘𝑦) = 𝑦)
109rexlimivw 3027 . . . . . . . 8 (∃𝑥𝐴 𝑦 = (card‘𝐵) → (card‘𝑦) = 𝑦)
115, 10sylbi 207 . . . . . . 7 (𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)} → (card‘𝑦) = 𝑦)
1211rgen 2922 . . . . . 6 𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)} (card‘𝑦) = 𝑦
13 carduni 8752 . . . . . 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 6160 . . . . . . 7 (card‘𝐵) ∈ V
1615dfiun2 4525 . . . . . 6 𝑥𝐴 (card‘𝐵) = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)}
1716fveq2i 6153 . . . . 5 (card‘ 𝑥𝐴 (card‘𝐵)) = (card‘ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (card‘𝐵)})
1814, 17, 163eqtr4g 2685 . . . 4 (𝐴𝑉 → (card‘ 𝑥𝐴 (card‘𝐵)) = 𝑥𝐴 (card‘𝐵))
1918adantr 481 . . 3 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝐵) = 𝐵) → (card‘ 𝑥𝐴 (card‘𝐵)) = 𝑥𝐴 (card‘𝐵))
20 iuneq2 4508 . . . . 5 (∀𝑥𝐴 (card‘𝐵) = 𝐵 𝑥𝐴 (card‘𝐵) = 𝑥𝐴 𝐵)
2120adantl 482 . . . 4 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝐵) = 𝐵) → 𝑥𝐴 (card‘𝐵) = 𝑥𝐴 𝐵)
2221fveq2d 6154 . . 3 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝐵) = 𝐵) → (card‘ 𝑥𝐴 (card‘𝐵)) = (card‘ 𝑥𝐴 𝐵))
2319, 22, 213eqtr3d 2668 . 2 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝐵) = 𝐵) → (card‘ 𝑥𝐴 𝐵) = 𝑥𝐴 𝐵)
2423ex 450 1 (𝐴𝑉 → (∀𝑥𝐴 (card‘𝐵) = 𝐵 → (card‘ 𝑥𝐴 𝐵) = 𝑥𝐴 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  {cab 2612  wral 2912  wrex 2913  Vcvv 3191   cuni 4407   ciun 4490  cfv 5850  cardccrd 8706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5688  df-on 5689  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-card 8710
This theorem is referenced by:  alephcard  8838
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