Theorem cardinfima 10189

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 3475	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
2		isinfcard 10184	. . . . . . . . . . . . 13 ⊢ ((ω ⊆ (𝐹‘𝑥) ∧ (card‘(𝐹‘𝑥)) = (𝐹‘𝑥)) ↔ (𝐹‘𝑥) ∈ ran ℵ)
3	2	bicomi 227	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑥) ∈ ran ℵ ↔ (ω ⊆ (𝐹‘𝑥) ∧ (card‘(𝐹‘𝑥)) = (𝐹‘𝑥)))
4	3	simplbi 502	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) ∈ ran ℵ → ω ⊆ (𝐹‘𝑥))
5		ffn 6720	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → 𝐹 Fn 𝐴)
6		fnfvelrn 7091	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ran 𝐹)
7	6	ex 418	. . . . . . . . . . . . . . 15 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ ran 𝐹))
8		fnima 6680	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)
9	8	eleq2d 2850	. . . . . . . . . . . . . . 15 ⊢ (𝐹 Fn 𝐴 → ((𝐹‘𝑥) ∈ (𝐹 “ 𝐴) ↔ (𝐹‘𝑥) ∈ ran 𝐹))
10	7, 9	sylibrd 262	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴)))
11		elssuni 4902	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑥) ∈ (𝐹 “ 𝐴) → (𝐹‘𝑥) ⊆ ∪ (𝐹 “ 𝐴))
12	10, 11	syl6 36	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ⊆ ∪ (𝐹 “ 𝐴)))
13	12	imp 412	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ⊆ ∪ (𝐹 “ 𝐴))
14	5, 13	sylan 592	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ⊆ ∪ (𝐹 “ 𝐴))
15	4, 14	sylan9ssr 3927	. . . . . . . . . 10 ⊢ (((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑥) ∈ ran ℵ) → ω ⊆ ∪ (𝐹 “ 𝐴))
16	15	anasss 472	. . . . . . . . 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 10188	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝐹:𝐴⟶(ω ∪ ran ℵ) → ∪ (𝐹 “ 𝐴) ∈ (ω ∪ ran ℵ)))
19		iscard3 10185	. . . . . . . . . 10 ⊢ ((card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴) ↔ ∪ (𝐹 “ 𝐴) ∈ (ω ∪ ran ℵ))
20	18, 19	imbitrrdi 255	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴)))
21	20	adantrd 497	. . . . . . . 8 ⊢ (𝐴 ∈ V → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ ran ℵ)) → (card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴)))
22	17, 21	jcad 522	. . . . . . 7 ⊢ (𝐴 ∈ V → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ ran ℵ)) → (ω ⊆ ∪ (𝐹 “ 𝐴) ∧ (card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴))))
23		isinfcard 10184	. . . . . . 7 ⊢ ((ω ⊆ ∪ (𝐹 “ 𝐴) ∧ (card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴)) ↔ ∪ (𝐹 “ 𝐴) ∈ ran ℵ)
24	22, 23	imbitrdi 254	. . . . . 6 ⊢ (𝐴 ∈ V → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ ran ℵ)) → ∪ (𝐹 “ 𝐴) ∈ ran ℵ))
25	24	exp4d 439	. . . . 5 ⊢ (𝐴 ∈ V → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) ∈ ran ℵ → ∪ (𝐹 “ 𝐴) ∈ ran ℵ))))
26	25	imp 412	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐹:𝐴⟶(ω ∪ ran ℵ)) → (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) ∈ ran ℵ → ∪ (𝐹 “ 𝐴) ∈ ran ℵ)))
27	26	rexlimdv 3165	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐹:𝐴⟶(ω ∪ ran ℵ)) → (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ ran ℵ → ∪ (𝐹 “ 𝐴) ∈ ran ℵ))
28	27	expimpd 459	. 2 ⊢ (𝐴 ∈ V → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ ran ℵ) → ∪ (𝐹 “ 𝐴) ∈ ran ℵ))
29	1, 28	syl 18	1 ⊢ (𝐴 ∈ 𝐵 → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ ran ℵ) → ∪ (𝐹 “ 𝐴) ∈ ran ℵ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∪ cun 4152  ∪ cuni 4870  ran crn 5656   “ cima 5658   Fn wfn 6542  ⟶wf 6543  ‘cfv 6547  ωcom 7887  cardccrd 10029  ℵcale 10030

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761  ax-inf2 9655

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  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 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6311  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-isom 6556  df-riota 7387  df-ov 7433  df-om 7888  df-2nd 8012  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-rdg 8432  df-1o 8490  df-er 8731  df-en 8988  df-dom 8989  df-sdom 8990  df-fin 8991  df-oi 9517  df-har 9564  df-card 10033  df-aleph 10034

This theorem is used by:  alephfplem4  10199