Theorem carduniima 10134

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 6740	. . . . 5 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → Fun 𝐹)
2		funimaexg 6654	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝐵) → (𝐹 “ 𝐴) ∈ V)
3	1, 2	sylan 580	. . . 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 6757	. . . . . . 7 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹 “ 𝐴) ⊆ (ω ∪ ran ℵ))
6	5	sseld 3994	. . . . . 6 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝑥 ∈ (𝐹 “ 𝐴) → 𝑥 ∈ (ω ∪ ran ℵ)))
7		iscard3 10131	. . . . . 6 ⊢ ((card‘𝑥) = 𝑥 ↔ 𝑥 ∈ (ω ∪ ran ℵ))
8	6, 7	imbitrrdi 252	. . . . 5 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝑥 ∈ (𝐹 “ 𝐴) → (card‘𝑥) = 𝑥))
9	8	ralrimiv 3143	. . . 4 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → ∀𝑥 ∈ (𝐹 “ 𝐴)(card‘𝑥) = 𝑥)
10		carduni 10019	. . . 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 10131	. 2 ⊢ ((card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴) ↔ ∪ (𝐹 “ 𝐴) ∈ (ω ∪ ran ℵ))
14	12, 13	imbitrdi 251	1 ⊢ (𝐴 ∈ 𝐵 → (𝐹:𝐴⟶(ω ∪ ran ℵ) → ∪ (𝐹 “ 𝐴) ∈ (ω ∪ ran ℵ)))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  ∀wral 3059  Vcvv 3478   ∪ cun 3961  ∪ cuni 4912  ran crn 5690   “ cima 5692  Fun wfun 6557  ⟶wf 6559  ‘cfv 6563  ωcom 7887  cardccrd 9973  ℵcale 9974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-oi 9548  df-har 9595  df-card 9977  df-aleph 9978

This theorem is referenced by:  cardinfima  10135