Theorem cardinfima 9784

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 3440	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
2		isinfcard 9779	. . . . . . . . . . . . 13 ⊢ ((ω ⊆ (𝐹‘𝑥) ∧ (card‘(𝐹‘𝑥)) = (𝐹‘𝑥)) ↔ (𝐹‘𝑥) ∈ ran ℵ)
3	2	bicomi 223	. . . . . . . . . . . 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 6584	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → 𝐹 Fn 𝐴)
6		fnfvelrn 6940	. . . . . . . . . . . . . . . 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 6547	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)
9	8	eleq2d 2824	. . . . . . . . . . . . . . 15 ⊢ (𝐹 Fn 𝐴 → ((𝐹‘𝑥) ∈ (𝐹 “ 𝐴) ↔ (𝐹‘𝑥) ∈ ran 𝐹))
10	7, 9	sylibrd 258	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴)))
11		elssuni 4868	. . . . . . . . . . . . . 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 579	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ⊆ ∪ (𝐹 “ 𝐴))
15	4, 14	sylan9ssr 3931	. . . . . . . . . 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 9783	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝐹:𝐴⟶(ω ∪ ran ℵ) → ∪ (𝐹 “ 𝐴) ∈ (ω ∪ ran ℵ)))
19		iscard3 9780	. . . . . . . . . 10 ⊢ ((card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴) ↔ ∪ (𝐹 “ 𝐴) ∈ (ω ∪ ran ℵ))
20	18, 19	syl6ibr 251	. . . . . . . . 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 9779	. . . . . . 7 ⊢ ((ω ⊆ ∪ (𝐹 “ 𝐴) ∧ (card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴)) ↔ ∪ (𝐹 “ 𝐴) ∈ ran ℵ)
24	22, 23	syl6ib 250	. . . . . 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 3211	. . 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 1539   ∈ wcel 2108  ∃wrex 3064  Vcvv 3422   ∪ cun 3881   ⊆ wss 3883  ∪ cuni 4836  ran crn 5581   “ cima 5583   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  ωcom 7687  cardccrd 9624  ℵcale 9625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-oi 9199  df-har 9246  df-card 9628  df-aleph 9629

This theorem is referenced by:  alephfplem4  9794