Theorem carduniima 10214

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 6742	. . . . 5 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → Fun 𝐹)
2		funimaexg 6656	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝐵) → (𝐹 “ 𝐴) ∈ V)
3	1, 2	sylan 592	. . . 4 ⊢ ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ 𝐴 ∈ 𝐵) → (𝐹 “ 𝐴) ∈ V)
4	3	expcom 419	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹 “ 𝐴) ∈ V))
5		fimass 6760	. . . . . . 7 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹 “ 𝐴) ⊆ (ω ∪ ran ℵ))
6	5	sseld 3912	. . . . . 6 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝑥 ∈ (𝐹 “ 𝐴) → 𝑥 ∈ (ω ∪ ran ℵ)))
7		iscard3 10211	. . . . . 6 ⊢ ((card‘𝑥) = 𝑥 ↔ 𝑥 ∈ (ω ∪ ran ℵ))
8	6, 7	imbitrrdi 255	. . . . 5 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝑥 ∈ (𝐹 “ 𝐴) → (card‘𝑥) = 𝑥))
9	8	ralrimiv 3157	. . . 4 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → ∀𝑥 ∈ (𝐹 “ 𝐴)(card‘𝑥) = 𝑥)
10		carduni 10101	. . . 4 ⊢ ((𝐹 “ 𝐴) ∈ V → (∀𝑥 ∈ (𝐹 “ 𝐴)(card‘𝑥) = 𝑥 → (card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴)))
11	9, 10	syl5 35	. . 3 ⊢ ((𝐹 “ 𝐴) ∈ V → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴)))
12	4, 11	syli 40	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴)))
13		iscard3 10211	. 2 ⊢ ((card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴) ↔ ∪ (𝐹 “ 𝐴) ∈ (ω ∪ ran ℵ))
14	12, 13	imbitrdi 254	1 ⊢ (𝐴 ∈ 𝐵 → (𝐹:𝐴⟶(ω ∪ ran ℵ) → ∪ (𝐹 “ 𝐴) ∈ (ω ∪ ran ℵ)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ∀wral 3080  Vcvv 3454   ∪ cun 4152  ∪ cuni 4872  ran crn 5659   “ cima 5661  Fun wfun 6560  ⟶wf 6562  ‘cfv 6566  ωcom 7911  cardccrd 10053  ℵcale 10054

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-inf2 9679

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-oi 9541  df-har 9588  df-card 10057  df-aleph 10058

This theorem is used by:  cardinfima  10215