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Theorem carduniima 9511
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 6514 . . . . 5 (𝐹:𝐴⟶(ω ∪ ran ℵ) → Fun 𝐹)
2 funimaexg 6437 . . . . 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 6552 . . . . . . 7 (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹𝐴) ⊆ (ω ∪ ran ℵ))
65sseld 3970 . . . . . 6 (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝑥 ∈ (𝐹𝐴) → 𝑥 ∈ (ω ∪ ran ℵ)))
7 iscard3 9508 . . . . . 6 ((card‘𝑥) = 𝑥𝑥 ∈ (ω ∪ ran ℵ))
86, 7syl6ibr 253 . . . . 5 (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝑥 ∈ (𝐹𝐴) → (card‘𝑥) = 𝑥))
98ralrimiv 3186 . . . 4 (𝐹:𝐴⟶(ω ∪ ran ℵ) → ∀𝑥 ∈ (𝐹𝐴)(card‘𝑥) = 𝑥)
10 carduni 9399 . . . 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 9508 . 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 1530  wcel 2107  wral 3143  Vcvv 3500  cun 3938   cuni 4837  ran crn 5555  cima 5557  Fun wfun 6346  wf 6348  cfv 6352  ωcom 7568  cardccrd 9353  cale 9354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-inf2 9093
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-isom 6361  df-riota 7106  df-om 7569  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-oi 8963  df-har 9011  df-card 9357  df-aleph 9358
This theorem is referenced by:  cardinfima  9512
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