Theorem cardinfima 9988

Description: If a mapping to cardinals has an infinite value, then the union of its image is an infinite cardinal. Corollary 11.17 of [TakeutiZaring] p. 104. (Contributed by NM, 4-Nov-2004.)

Assertion

Ref	Expression
cardinfima	⊢ (𝐴 ∈ 𝐵 → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ ran ℵ) → ∪ (𝐹 “ 𝐴) ∈ ran ℵ))

Distinct variable groups:   𝑥,𝐹   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem cardinfima

Step	Hyp	Ref	Expression
1		elex 3457	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
2		isinfcard 9983	. . . . . . . . . . . . 13 ⊢ ((ω ⊆ (𝐹‘𝑥) ∧ (card‘(𝐹‘𝑥)) = (𝐹‘𝑥)) ↔ (𝐹‘𝑥) ∈ ran ℵ)
3	2	bicomi 224	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑥) ∈ ran ℵ ↔ (ω ⊆ (𝐹‘𝑥) ∧ (card‘(𝐹‘𝑥)) = (𝐹‘𝑥)))
4	3	simplbi 497	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) ∈ ran ℵ → ω ⊆ (𝐹‘𝑥))
5		ffn 6651	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → 𝐹 Fn 𝐴)
6		fnfvelrn 7013	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ran 𝐹)
7	6	ex 412	. . . . . . . . . . . . . . 15 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ ran 𝐹))
8		fnima 6611	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)
9	8	eleq2d 2817	. . . . . . . . . . . . . . 15 ⊢ (𝐹 Fn 𝐴 → ((𝐹‘𝑥) ∈ (𝐹 “ 𝐴) ↔ (𝐹‘𝑥) ∈ ran 𝐹))
10	7, 9	sylibrd 259	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴)))
11		elssuni 4887	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑥) ∈ (𝐹 “ 𝐴) → (𝐹‘𝑥) ⊆ ∪ (𝐹 “ 𝐴))
12	10, 11	syl6 35	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ⊆ ∪ (𝐹 “ 𝐴)))
13	12	imp 406	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ⊆ ∪ (𝐹 “ 𝐴))
14	5, 13	sylan 580	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ⊆ ∪ (𝐹 “ 𝐴))
15	4, 14	sylan9ssr 3944	. . . . . . . . . 10 ⊢ (((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑥) ∈ ran ℵ) → ω ⊆ ∪ (𝐹 “ 𝐴))
16	15	anasss 466	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ ran ℵ)) → ω ⊆ ∪ (𝐹 “ 𝐴))
17	16	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ V → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ ran ℵ)) → ω ⊆ ∪ (𝐹 “ 𝐴)))
18		carduniima 9987	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝐹:𝐴⟶(ω ∪ ran ℵ) → ∪ (𝐹 “ 𝐴) ∈ (ω ∪ ran ℵ)))
19		iscard3 9984	. . . . . . . . . 10 ⊢ ((card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴) ↔ ∪ (𝐹 “ 𝐴) ∈ (ω ∪ ran ℵ))
20	18, 19	imbitrrdi 252	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴)))
21	20	adantrd 491	. . . . . . . 8 ⊢ (𝐴 ∈ V → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ ran ℵ)) → (card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴)))
22	17, 21	jcad 512	. . . . . . 7 ⊢ (𝐴 ∈ V → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ ran ℵ)) → (ω ⊆ ∪ (𝐹 “ 𝐴) ∧ (card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴))))
23		isinfcard 9983	. . . . . . 7 ⊢ ((ω ⊆ ∪ (𝐹 “ 𝐴) ∧ (card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴)) ↔ ∪ (𝐹 “ 𝐴) ∈ ran ℵ)
24	22, 23	imbitrdi 251	. . . . . 6 ⊢ (𝐴 ∈ V → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ ran ℵ)) → ∪ (𝐹 “ 𝐴) ∈ ran ℵ))
25	24	exp4d 433	. . . . 5 ⊢ (𝐴 ∈ V → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) ∈ ran ℵ → ∪ (𝐹 “ 𝐴) ∈ ran ℵ))))
26	25	imp 406	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐹:𝐴⟶(ω ∪ ran ℵ)) → (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) ∈ ran ℵ → ∪ (𝐹 “ 𝐴) ∈ ran ℵ)))
27	26	rexlimdv 3131	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐹:𝐴⟶(ω ∪ ran ℵ)) → (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ ran ℵ → ∪ (𝐹 “ 𝐴) ∈ ran ℵ))
28	27	expimpd 453	. 2 ⊢ (𝐴 ∈ V → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ ran ℵ) → ∪ (𝐹 “ 𝐴) ∈ ran ℵ))
29	1, 28	syl 17	1 ⊢ (𝐴 ∈ 𝐵 → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ ran ℵ) → ∪ (𝐹 “ 𝐴) ∈ ran ℵ))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  Vcvv 3436   ∪ cun 3895   ⊆ wss 3897  ∪ cuni 4856  ran crn 5615   “ cima 5617   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  ωcom 7796  cardccrd 9828  ℵcale 9829

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-inf2 9531

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-oi 9396  df-har 9443  df-card 9832  df-aleph 9833

This theorem is referenced by:  alephfplem4  9998