Theorem cardinfima 10122

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 3480	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
2		isinfcard 10117	. . . . . . . . . . . . 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 496	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) ∈ ran ℵ → ω ⊆ (𝐹‘𝑥))
5		ffn 6723	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → 𝐹 Fn 𝐴)
6		fnfvelrn 7089	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ran 𝐹)
7	6	ex 411	. . . . . . . . . . . . . . 15 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ ran 𝐹))
8		fnima 6686	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)
9	8	eleq2d 2811	. . . . . . . . . . . . . . 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 4941	. . . . . . . . . . . . . 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 405	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ⊆ ∪ (𝐹 “ 𝐴))
14	5, 13	sylan 578	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ⊆ ∪ (𝐹 “ 𝐴))
15	4, 14	sylan9ssr 3991	. . . . . . . . . 10 ⊢ (((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑥) ∈ ran ℵ) → ω ⊆ ∪ (𝐹 “ 𝐴))
16	15	anasss 465	. . . . . . . . 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 10121	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝐹:𝐴⟶(ω ∪ ran ℵ) → ∪ (𝐹 “ 𝐴) ∈ (ω ∪ ran ℵ)))
19		iscard3 10118	. . . . . . . . . 10 ⊢ ((card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴) ↔ ∪ (𝐹 “ 𝐴) ∈ (ω ∪ ran ℵ))
20	18, 19	imbitrrdi 251	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴)))
21	20	adantrd 490	. . . . . . . 8 ⊢ (𝐴 ∈ V → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ ran ℵ)) → (card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴)))
22	17, 21	jcad 511	. . . . . . 7 ⊢ (𝐴 ∈ V → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ ran ℵ)) → (ω ⊆ ∪ (𝐹 “ 𝐴) ∧ (card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴))))
23		isinfcard 10117	. . . . . . 7 ⊢ ((ω ⊆ ∪ (𝐹 “ 𝐴) ∧ (card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴)) ↔ ∪ (𝐹 “ 𝐴) ∈ ran ℵ)
24	22, 23	imbitrdi 250	. . . . . 6 ⊢ (𝐴 ∈ V → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ ran ℵ)) → ∪ (𝐹 “ 𝐴) ∈ ran ℵ))
25	24	exp4d 432	. . . . 5 ⊢ (𝐴 ∈ V → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) ∈ ran ℵ → ∪ (𝐹 “ 𝐴) ∈ ran ℵ))))
26	25	imp 405	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐹:𝐴⟶(ω ∪ ran ℵ)) → (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) ∈ ran ℵ → ∪ (𝐹 “ 𝐴) ∈ ran ℵ)))
27	26	rexlimdv 3142	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐹:𝐴⟶(ω ∪ ran ℵ)) → (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ ran ℵ → ∪ (𝐹 “ 𝐴) ∈ ran ℵ))
28	27	expimpd 452	. 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 394   = wceq 1533   ∈ wcel 2098  ∃wrex 3059  Vcvv 3461   ∪ cun 3942   ⊆ wss 3944  ∪ cuni 4909  ran crn 5679   “ cima 5681   Fn wfn 6544  ⟶wf 6545  ‘cfv 6549  ωcom 7871  cardccrd 9960  ℵcale 9961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-oi 9535  df-har 9582  df-card 9964  df-aleph 9965

This theorem is referenced by:  alephfplem4  10132