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Theorem cardinfima 9206
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 (𝐴𝐵 → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ ∃𝑥𝐴 (𝐹𝑥) ∈ ran ℵ) → (𝐹𝐴) ∈ ran ℵ))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem cardinfima
StepHypRef Expression
1 elex 3400 . 2 (𝐴𝐵𝐴 ∈ V)
2 isinfcard 9201 . . . . . . . . . . . . 13 ((ω ⊆ (𝐹𝑥) ∧ (card‘(𝐹𝑥)) = (𝐹𝑥)) ↔ (𝐹𝑥) ∈ ran ℵ)
32bicomi 216 . . . . . . . . . . . 12 ((𝐹𝑥) ∈ ran ℵ ↔ (ω ⊆ (𝐹𝑥) ∧ (card‘(𝐹𝑥)) = (𝐹𝑥)))
43simplbi 492 . . . . . . . . . . 11 ((𝐹𝑥) ∈ ran ℵ → ω ⊆ (𝐹𝑥))
5 ffn 6256 . . . . . . . . . . . 12 (𝐹:𝐴⟶(ω ∪ ran ℵ) → 𝐹 Fn 𝐴)
6 fnfvelrn 6582 . . . . . . . . . . . . . . . 16 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹𝑥) ∈ ran 𝐹)
76ex 402 . . . . . . . . . . . . . . 15 (𝐹 Fn 𝐴 → (𝑥𝐴 → (𝐹𝑥) ∈ ran 𝐹))
8 fnima 6221 . . . . . . . . . . . . . . . 16 (𝐹 Fn 𝐴 → (𝐹𝐴) = ran 𝐹)
98eleq2d 2864 . . . . . . . . . . . . . . 15 (𝐹 Fn 𝐴 → ((𝐹𝑥) ∈ (𝐹𝐴) ↔ (𝐹𝑥) ∈ ran 𝐹))
107, 9sylibrd 251 . . . . . . . . . . . . . 14 (𝐹 Fn 𝐴 → (𝑥𝐴 → (𝐹𝑥) ∈ (𝐹𝐴)))
11 elssuni 4659 . . . . . . . . . . . . . 14 ((𝐹𝑥) ∈ (𝐹𝐴) → (𝐹𝑥) ⊆ (𝐹𝐴))
1210, 11syl6 35 . . . . . . . . . . . . 13 (𝐹 Fn 𝐴 → (𝑥𝐴 → (𝐹𝑥) ⊆ (𝐹𝐴)))
1312imp 396 . . . . . . . . . . . 12 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹𝑥) ⊆ (𝐹𝐴))
145, 13sylan 576 . . . . . . . . . . 11 ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ 𝑥𝐴) → (𝐹𝑥) ⊆ (𝐹𝐴))
154, 14sylan9ssr 3812 . . . . . . . . . 10 (((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ 𝑥𝐴) ∧ (𝐹𝑥) ∈ ran ℵ) → ω ⊆ (𝐹𝐴))
1615anasss 459 . . . . . . . . 9 ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ (𝑥𝐴 ∧ (𝐹𝑥) ∈ ran ℵ)) → ω ⊆ (𝐹𝐴))
1716a1i 11 . . . . . . . 8 (𝐴 ∈ V → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ (𝑥𝐴 ∧ (𝐹𝑥) ∈ ran ℵ)) → ω ⊆ (𝐹𝐴)))
18 carduniima 9205 . . . . . . . . . 10 (𝐴 ∈ V → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹𝐴) ∈ (ω ∪ ran ℵ)))
19 iscard3 9202 . . . . . . . . . 10 ((card‘ (𝐹𝐴)) = (𝐹𝐴) ↔ (𝐹𝐴) ∈ (ω ∪ ran ℵ))
2018, 19syl6ibr 244 . . . . . . . . 9 (𝐴 ∈ V → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (card‘ (𝐹𝐴)) = (𝐹𝐴)))
2120adantrd 486 . . . . . . . 8 (𝐴 ∈ V → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ (𝑥𝐴 ∧ (𝐹𝑥) ∈ ran ℵ)) → (card‘ (𝐹𝐴)) = (𝐹𝐴)))
2217, 21jcad 509 . . . . . . 7 (𝐴 ∈ V → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ (𝑥𝐴 ∧ (𝐹𝑥) ∈ ran ℵ)) → (ω ⊆ (𝐹𝐴) ∧ (card‘ (𝐹𝐴)) = (𝐹𝐴))))
23 isinfcard 9201 . . . . . . 7 ((ω ⊆ (𝐹𝐴) ∧ (card‘ (𝐹𝐴)) = (𝐹𝐴)) ↔ (𝐹𝐴) ∈ ran ℵ)
2422, 23syl6ib 243 . . . . . 6 (𝐴 ∈ V → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ (𝑥𝐴 ∧ (𝐹𝑥) ∈ ran ℵ)) → (𝐹𝐴) ∈ ran ℵ))
2524exp4d 425 . . . . 5 (𝐴 ∈ V → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝑥𝐴 → ((𝐹𝑥) ∈ ran ℵ → (𝐹𝐴) ∈ ran ℵ))))
2625imp 396 . . . 4 ((𝐴 ∈ V ∧ 𝐹:𝐴⟶(ω ∪ ran ℵ)) → (𝑥𝐴 → ((𝐹𝑥) ∈ ran ℵ → (𝐹𝐴) ∈ ran ℵ)))
2726rexlimdv 3211 . . 3 ((𝐴 ∈ V ∧ 𝐹:𝐴⟶(ω ∪ ran ℵ)) → (∃𝑥𝐴 (𝐹𝑥) ∈ ran ℵ → (𝐹𝐴) ∈ ran ℵ))
2827expimpd 446 . 2 (𝐴 ∈ V → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ ∃𝑥𝐴 (𝐹𝑥) ∈ ran ℵ) → (𝐹𝐴) ∈ ran ℵ))
291, 28syl 17 1 (𝐴𝐵 → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ ∃𝑥𝐴 (𝐹𝑥) ∈ ran ℵ) → (𝐹𝐴) ∈ ran ℵ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wrex 3090  Vcvv 3385  cun 3767  wss 3769   cuni 4628  ran crn 5313  cima 5315   Fn wfn 6096  wf 6097  cfv 6101  ωcom 7299  cardccrd 9047  cale 9048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-inf2 8788
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-se 5272  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-riota 6839  df-om 7300  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-oi 8657  df-har 8705  df-card 9051  df-aleph 9052
This theorem is referenced by:  alephfplem4  9216
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