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Theorem carduniima 9200
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 6257 . . . . 5 (𝐹:𝐴⟶(ω ∪ ran ℵ) → Fun 𝐹)
2 funimaexg 6184 . . . . 5 ((Fun 𝐹𝐴𝐵) → (𝐹𝐴) ∈ V)
31, 2sylan 571 . . . 4 ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ 𝐴𝐵) → (𝐹𝐴) ∈ V)
43expcom 400 . . 3 (𝐴𝐵 → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹𝐴) ∈ V))
5 ffn 6254 . . . . . . . . 9 (𝐹:𝐴⟶(ω ∪ ran ℵ) → 𝐹 Fn 𝐴)
6 fnima 6219 . . . . . . . . 9 (𝐹 Fn 𝐴 → (𝐹𝐴) = ran 𝐹)
75, 6syl 17 . . . . . . . 8 (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹𝐴) = ran 𝐹)
8 frn 6260 . . . . . . . 8 (𝐹:𝐴⟶(ω ∪ ran ℵ) → ran 𝐹 ⊆ (ω ∪ ran ℵ))
97, 8eqsstrd 3834 . . . . . . 7 (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹𝐴) ⊆ (ω ∪ ran ℵ))
109sseld 3795 . . . . . 6 (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝑥 ∈ (𝐹𝐴) → 𝑥 ∈ (ω ∪ ran ℵ)))
11 iscard3 9197 . . . . . 6 ((card‘𝑥) = 𝑥𝑥 ∈ (ω ∪ ran ℵ))
1210, 11syl6ibr 243 . . . . 5 (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝑥 ∈ (𝐹𝐴) → (card‘𝑥) = 𝑥))
1312ralrimiv 3151 . . . 4 (𝐹:𝐴⟶(ω ∪ ran ℵ) → ∀𝑥 ∈ (𝐹𝐴)(card‘𝑥) = 𝑥)
14 carduni 9088 . . . 4 ((𝐹𝐴) ∈ V → (∀𝑥 ∈ (𝐹𝐴)(card‘𝑥) = 𝑥 → (card‘ (𝐹𝐴)) = (𝐹𝐴)))
1513, 14syl5 34 . . 3 ((𝐹𝐴) ∈ V → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (card‘ (𝐹𝐴)) = (𝐹𝐴)))
164, 15syli 39 . 2 (𝐴𝐵 → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (card‘ (𝐹𝐴)) = (𝐹𝐴)))
17 iscard3 9197 . 2 ((card‘ (𝐹𝐴)) = (𝐹𝐴) ↔ (𝐹𝐴) ∈ (ω ∪ ran ℵ))
1816, 17syl6ib 242 1 (𝐴𝐵 → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹𝐴) ∈ (ω ∪ ran ℵ)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1637  wcel 2156  wral 3094  Vcvv 3389  cun 3765   cuni 4628  ran crn 5310  cima 5312  Fun wfun 6093   Fn wfn 6094  wf 6095  cfv 6099  ωcom 7293  cardccrd 9042  cale 9043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2782  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5094  ax-un 7177  ax-inf2 8783
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2791  df-cleq 2797  df-clel 2800  df-nfc 2935  df-ne 2977  df-ral 3099  df-rex 3100  df-reu 3101  df-rmo 3102  df-rab 3103  df-v 3391  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4115  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4843  df-opab 4905  df-mpt 4922  df-tr 4945  df-id 5217  df-eprel 5222  df-po 5230  df-so 5231  df-fr 5268  df-se 5269  df-we 5270  df-xp 5315  df-rel 5316  df-cnv 5317  df-co 5318  df-dm 5319  df-rn 5320  df-res 5321  df-ima 5322  df-pred 5891  df-ord 5937  df-on 5938  df-lim 5939  df-suc 5940  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-isom 6108  df-riota 6833  df-om 7294  df-wrecs 7640  df-recs 7702  df-rdg 7740  df-er 7977  df-en 8191  df-dom 8192  df-sdom 8193  df-fin 8194  df-oi 8652  df-har 8700  df-card 9046  df-aleph 9047
This theorem is referenced by:  cardinfima  9201
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