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Theorem carduniima 8957
 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 6086 . . . . 5 (𝐹:𝐴⟶(ω ∪ ran ℵ) → Fun 𝐹)
2 funimaexg 6013 . . . . 5 ((Fun 𝐹𝐴𝐵) → (𝐹𝐴) ∈ V)
31, 2sylan 487 . . . 4 ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ 𝐴𝐵) → (𝐹𝐴) ∈ V)
43expcom 450 . . 3 (𝐴𝐵 → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹𝐴) ∈ V))
5 ffn 6083 . . . . . . . . 9 (𝐹:𝐴⟶(ω ∪ ran ℵ) → 𝐹 Fn 𝐴)
6 fnima 6048 . . . . . . . . 9 (𝐹 Fn 𝐴 → (𝐹𝐴) = ran 𝐹)
75, 6syl 17 . . . . . . . 8 (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹𝐴) = ran 𝐹)
8 frn 6091 . . . . . . . 8 (𝐹:𝐴⟶(ω ∪ ran ℵ) → ran 𝐹 ⊆ (ω ∪ ran ℵ))
97, 8eqsstrd 3672 . . . . . . 7 (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹𝐴) ⊆ (ω ∪ ran ℵ))
109sseld 3635 . . . . . 6 (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝑥 ∈ (𝐹𝐴) → 𝑥 ∈ (ω ∪ ran ℵ)))
11 iscard3 8954 . . . . . 6 ((card‘𝑥) = 𝑥𝑥 ∈ (ω ∪ ran ℵ))
1210, 11syl6ibr 242 . . . . 5 (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝑥 ∈ (𝐹𝐴) → (card‘𝑥) = 𝑥))
1312ralrimiv 2994 . . . 4 (𝐹:𝐴⟶(ω ∪ ran ℵ) → ∀𝑥 ∈ (𝐹𝐴)(card‘𝑥) = 𝑥)
14 carduni 8845 . . . 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 8954 . 2 ((card‘ (𝐹𝐴)) = (𝐹𝐴) ↔ (𝐹𝐴) ∈ (ω ∪ ran ℵ))
1816, 17syl6ib 241 1 (𝐴𝐵 → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹𝐴) ∈ (ω ∪ ran ℵ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030  ∀wral 2941  Vcvv 3231   ∪ cun 3605  ∪ cuni 4468  ran crn 5144   “ cima 5146  Fun wfun 5920   Fn wfn 5921  ⟶wf 5922  ‘cfv 5926  ωcom 7107  cardccrd 8799  ℵcale 8800 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  ax-inf2 8576 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-rmo 2949  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-se 5103  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-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  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-isom 5935  df-riota 6651  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-oi 8456  df-har 8504  df-card 8803  df-aleph 8804 This theorem is referenced by:  cardinfima  8958
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