Theorem cardinfima 10137

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 3501	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
2		isinfcard 10132	. . . . . . . . . . . . 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 6736	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → 𝐹 Fn 𝐴)
6		fnfvelrn 7100	. . . . . . . . . . . . . . . 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 6698	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)
9	8	eleq2d 2827	. . . . . . . . . . . . . . 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 4937	. . . . . . . . . . . . . 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 3998	. . . . . . . . . 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 10136	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝐹:𝐴⟶(ω ∪ ran ℵ) → ∪ (𝐹 “ 𝐴) ∈ (ω ∪ ran ℵ)))
19		iscard3 10133	. . . . . . . . . 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 10132	. . . . . . 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 3153	. . 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 1540   ∈ wcel 2108  ∃wrex 3070  Vcvv 3480   ∪ cun 3949   ⊆ wss 3951  ∪ cuni 4907  ran crn 5686   “ cima 5688   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  ωcom 7887  cardccrd 9975  ℵcale 9976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-oi 9550  df-har 9597  df-card 9979  df-aleph 9980

This theorem is referenced by:  alephfplem4  10147