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Theorem carduniima 9510
Description: The union of the image of a mapping to cardinals is a cardinal. Proposition 11.16 of [TakeutiZaring] p. 104. (Contributed by NM, 4-Nov-2004.)
Assertion
Ref Expression
carduniima (𝐴𝐵 → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹𝐴) ∈ (ω ∪ ran ℵ)))

Proof of Theorem carduniima
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ffun 6510 . . . . 5 (𝐹:𝐴⟶(ω ∪ ran ℵ) → Fun 𝐹)
2 funimaexg 6433 . . . . 5 ((Fun 𝐹𝐴𝐵) → (𝐹𝐴) ∈ V)
31, 2sylan 580 . . . 4 ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ 𝐴𝐵) → (𝐹𝐴) ∈ V)
43expcom 414 . . 3 (𝐴𝐵 → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹𝐴) ∈ V))
5 fimass 6548 . . . . . . 7 (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹𝐴) ⊆ (ω ∪ ran ℵ))
65sseld 3963 . . . . . 6 (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝑥 ∈ (𝐹𝐴) → 𝑥 ∈ (ω ∪ ran ℵ)))
7 iscard3 9507 . . . . . 6 ((card‘𝑥) = 𝑥𝑥 ∈ (ω ∪ ran ℵ))
86, 7syl6ibr 253 . . . . 5 (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝑥 ∈ (𝐹𝐴) → (card‘𝑥) = 𝑥))
98ralrimiv 3178 . . . 4 (𝐹:𝐴⟶(ω ∪ ran ℵ) → ∀𝑥 ∈ (𝐹𝐴)(card‘𝑥) = 𝑥)
10 carduni 9398 . . . 4 ((𝐹𝐴) ∈ V → (∀𝑥 ∈ (𝐹𝐴)(card‘𝑥) = 𝑥 → (card‘ (𝐹𝐴)) = (𝐹𝐴)))
119, 10syl5 34 . . 3 ((𝐹𝐴) ∈ V → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (card‘ (𝐹𝐴)) = (𝐹𝐴)))
124, 11syli 39 . 2 (𝐴𝐵 → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (card‘ (𝐹𝐴)) = (𝐹𝐴)))
13 iscard3 9507 . 2 ((card‘ (𝐹𝐴)) = (𝐹𝐴) ↔ (𝐹𝐴) ∈ (ω ∪ ran ℵ))
1412, 13syl6ib 252 1 (𝐴𝐵 → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹𝐴) ∈ (ω ∪ ran ℵ)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  cun 3931   cuni 4830  ran crn 5549  cima 5551  Fun wfun 6342  wf 6344  cfv 6348  ωcom 7569  cardccrd 9352  cale 9353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-oi 8962  df-har 9010  df-card 9356  df-aleph 9357
This theorem is referenced by:  cardinfima  9511
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