Theorem carduniima 9522

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	⊢ (𝐴 ∈ 𝐵 → (𝐹:𝐴⟶(ω ∪ ran ℵ) → ∪ (𝐹 “ 𝐴) ∈ (ω ∪ ran ℵ)))

Proof of Theorem carduniima

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffun 6517	. . . . 5 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → Fun 𝐹)
2		funimaexg 6440	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝐵) → (𝐹 “ 𝐴) ∈ V)
3	1, 2	sylan 582	. . . 4 ⊢ ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ 𝐴 ∈ 𝐵) → (𝐹 “ 𝐴) ∈ V)
4	3	expcom 416	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹 “ 𝐴) ∈ V))
5		fimass 6555	. . . . . . 7 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹 “ 𝐴) ⊆ (ω ∪ ran ℵ))
6	5	sseld 3966	. . . . . 6 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝑥 ∈ (𝐹 “ 𝐴) → 𝑥 ∈ (ω ∪ ran ℵ)))
7		iscard3 9519	. . . . . 6 ⊢ ((card‘𝑥) = 𝑥 ↔ 𝑥 ∈ (ω ∪ ran ℵ))
8	6, 7	syl6ibr 254	. . . . 5 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝑥 ∈ (𝐹 “ 𝐴) → (card‘𝑥) = 𝑥))
9	8	ralrimiv 3181	. . . 4 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → ∀𝑥 ∈ (𝐹 “ 𝐴)(card‘𝑥) = 𝑥)
10		carduni 9410	. . . 4 ⊢ ((𝐹 “ 𝐴) ∈ V → (∀𝑥 ∈ (𝐹 “ 𝐴)(card‘𝑥) = 𝑥 → (card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴)))
11	9, 10	syl5 34	. . 3 ⊢ ((𝐹 “ 𝐴) ∈ V → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴)))
12	4, 11	syli 39	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴)))
13		iscard3 9519	. 2 ⊢ ((card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴) ↔ ∪ (𝐹 “ 𝐴) ∈ (ω ∪ ran ℵ))
14	12, 13	syl6ib 253	1 ⊢ (𝐴 ∈ 𝐵 → (𝐹:𝐴⟶(ω ∪ ran ℵ) → ∪ (𝐹 “ 𝐴) ∈ (ω ∪ ran ℵ)))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  ∀wral 3138  Vcvv 3494   ∪ cun 3934  ∪ cuni 4838  ran crn 5556   “ cima 5558  Fun wfun 6349  ⟶wf 6351  ‘cfv 6355  ωcom 7580  cardccrd 9364  ℵcale 9365

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-inf2 9104

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-om 7581  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-oi 8974  df-har 9022  df-card 9368  df-aleph 9369

This theorem is referenced by:  cardinfima  9523