Theorem carduniima 10094

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 6721	. . . . 5 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → Fun 𝐹)
2		funimaexg 6635	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝐵) → (𝐹 “ 𝐴) ∈ V)
3	1, 2	sylan 579	. . . 4 ⊢ ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ 𝐴 ∈ 𝐵) → (𝐹 “ 𝐴) ∈ V)
4	3	expcom 413	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹 “ 𝐴) ∈ V))
5		fimass 6739	. . . . . . 7 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹 “ 𝐴) ⊆ (ω ∪ ran ℵ))
6	5	sseld 3982	. . . . . 6 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝑥 ∈ (𝐹 “ 𝐴) → 𝑥 ∈ (ω ∪ ran ℵ)))
7		iscard3 10091	. . . . . 6 ⊢ ((card‘𝑥) = 𝑥 ↔ 𝑥 ∈ (ω ∪ ran ℵ))
8	6, 7	imbitrrdi 251	. . . . 5 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝑥 ∈ (𝐹 “ 𝐴) → (card‘𝑥) = 𝑥))
9	8	ralrimiv 3144	. . . 4 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → ∀𝑥 ∈ (𝐹 “ 𝐴)(card‘𝑥) = 𝑥)
10		carduni 9979	. . . 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 10091	. 2 ⊢ ((card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴) ↔ ∪ (𝐹 “ 𝐴) ∈ (ω ∪ ran ℵ))
14	12, 13	imbitrdi 250	1 ⊢ (𝐴 ∈ 𝐵 → (𝐹:𝐴⟶(ω ∪ ran ℵ) → ∪ (𝐹 “ 𝐴) ∈ (ω ∪ ran ℵ)))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105  ∀wral 3060  Vcvv 3473   ∪ cun 3947  ∪ cuni 4909  ran crn 5678   “ cima 5680  Fun wfun 6538  ⟶wf 6540  ‘cfv 6544  ωcom 7858  cardccrd 9933  ℵcale 9934

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-inf2 9639

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7368  df-ov 7415  df-om 7859  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-er 8706  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-oi 9508  df-har 9555  df-card 9937  df-aleph 9938

This theorem is referenced by:  cardinfima  10095