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Theorem carduniima 8774
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 5942 . . . . 5 (𝐹:𝐴⟶(ω ∪ ran ℵ) → Fun 𝐹)
2 funimaexg 5870 . . . . 5 ((Fun 𝐹𝐴𝐵) → (𝐹𝐴) ∈ V)
31, 2sylan 486 . . . 4 ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ 𝐴𝐵) → (𝐹𝐴) ∈ V)
43expcom 449 . . 3 (𝐴𝐵 → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹𝐴) ∈ V))
5 ffn 5939 . . . . . . . . 9 (𝐹:𝐴⟶(ω ∪ ran ℵ) → 𝐹 Fn 𝐴)
6 fnima 5904 . . . . . . . . 9 (𝐹 Fn 𝐴 → (𝐹𝐴) = ran 𝐹)
75, 6syl 17 . . . . . . . 8 (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹𝐴) = ran 𝐹)
8 frn 5947 . . . . . . . 8 (𝐹:𝐴⟶(ω ∪ ran ℵ) → ran 𝐹 ⊆ (ω ∪ ran ℵ))
97, 8eqsstrd 3596 . . . . . . 7 (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹𝐴) ⊆ (ω ∪ ran ℵ))
109sseld 3561 . . . . . 6 (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝑥 ∈ (𝐹𝐴) → 𝑥 ∈ (ω ∪ ran ℵ)))
11 iscard3 8771 . . . . . 6 ((card‘𝑥) = 𝑥𝑥 ∈ (ω ∪ ran ℵ))
1210, 11syl6ibr 240 . . . . 5 (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝑥 ∈ (𝐹𝐴) → (card‘𝑥) = 𝑥))
1312ralrimiv 2942 . . . 4 (𝐹:𝐴⟶(ω ∪ ran ℵ) → ∀𝑥 ∈ (𝐹𝐴)(card‘𝑥) = 𝑥)
14 carduni 8662 . . . 4 ((𝐹𝐴) ∈ V → (∀𝑥 ∈ (𝐹𝐴)(card‘𝑥) = 𝑥 → (card‘ (𝐹𝐴)) = (𝐹𝐴)))
1513, 14syl5 33 . . 3 ((𝐹𝐴) ∈ V → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (card‘ (𝐹𝐴)) = (𝐹𝐴)))
164, 15syli 38 . 2 (𝐴𝐵 → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (card‘ (𝐹𝐴)) = (𝐹𝐴)))
17 iscard3 8771 . 2 ((card‘ (𝐹𝐴)) = (𝐹𝐴) ↔ (𝐹𝐴) ∈ (ω ∪ ran ℵ))
1816, 17syl6ib 239 1 (𝐴𝐵 → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹𝐴) ∈ (ω ∪ ran ℵ)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1975  wral 2890  Vcvv 3167  cun 3532   cuni 4361  ran crn 5024  cima 5026  Fun wfun 5779   Fn wfn 5780  wf 5781  cfv 5785  ωcom 6929  cardccrd 8616  cale 8617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819  ax-inf2 8393
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-reu 2897  df-rmo 2898  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-pss 3550  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-tp 4124  df-op 4126  df-uni 4362  df-int 4400  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-tr 4670  df-eprel 4934  df-id 4938  df-po 4944  df-so 4945  df-fr 4982  df-se 4983  df-we 4984  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-pred 5578  df-ord 5624  df-on 5625  df-lim 5626  df-suc 5627  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-isom 5794  df-riota 6484  df-om 6930  df-wrecs 7266  df-recs 7327  df-rdg 7365  df-er 7601  df-en 7814  df-dom 7815  df-sdom 7816  df-fin 7817  df-oi 8270  df-har 8318  df-card 8620  df-aleph 8621
This theorem is referenced by:  cardinfima  8775
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