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Theorem cardinfima 8776
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 3180 . 2 (𝐴𝐵𝐴 ∈ V)
2 isinfcard 8771 . . . . . . . . . . . . 13 ((ω ⊆ (𝐹𝑥) ∧ (card‘(𝐹𝑥)) = (𝐹𝑥)) ↔ (𝐹𝑥) ∈ ran ℵ)
32bicomi 212 . . . . . . . . . . . 12 ((𝐹𝑥) ∈ ran ℵ ↔ (ω ⊆ (𝐹𝑥) ∧ (card‘(𝐹𝑥)) = (𝐹𝑥)))
43simplbi 474 . . . . . . . . . . 11 ((𝐹𝑥) ∈ ran ℵ → ω ⊆ (𝐹𝑥))
5 ffn 5940 . . . . . . . . . . . 12 (𝐹:𝐴⟶(ω ∪ ran ℵ) → 𝐹 Fn 𝐴)
6 fnfvelrn 6245 . . . . . . . . . . . . . . . 16 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹𝑥) ∈ ran 𝐹)
76ex 448 . . . . . . . . . . . . . . 15 (𝐹 Fn 𝐴 → (𝑥𝐴 → (𝐹𝑥) ∈ ran 𝐹))
8 fnima 5905 . . . . . . . . . . . . . . . 16 (𝐹 Fn 𝐴 → (𝐹𝐴) = ran 𝐹)
98eleq2d 2668 . . . . . . . . . . . . . . 15 (𝐹 Fn 𝐴 → ((𝐹𝑥) ∈ (𝐹𝐴) ↔ (𝐹𝑥) ∈ ran 𝐹))
107, 9sylibrd 247 . . . . . . . . . . . . . 14 (𝐹 Fn 𝐴 → (𝑥𝐴 → (𝐹𝑥) ∈ (𝐹𝐴)))
11 elssuni 4393 . . . . . . . . . . . . . 14 ((𝐹𝑥) ∈ (𝐹𝐴) → (𝐹𝑥) ⊆ (𝐹𝐴))
1210, 11syl6 34 . . . . . . . . . . . . 13 (𝐹 Fn 𝐴 → (𝑥𝐴 → (𝐹𝑥) ⊆ (𝐹𝐴)))
1312imp 443 . . . . . . . . . . . 12 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹𝑥) ⊆ (𝐹𝐴))
145, 13sylan 486 . . . . . . . . . . 11 ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ 𝑥𝐴) → (𝐹𝑥) ⊆ (𝐹𝐴))
154, 14sylan9ssr 3577 . . . . . . . . . 10 (((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ 𝑥𝐴) ∧ (𝐹𝑥) ∈ ran ℵ) → ω ⊆ (𝐹𝐴))
1615anasss 676 . . . . . . . . 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 8775 . . . . . . . . . 10 (𝐴 ∈ V → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹𝐴) ∈ (ω ∪ ran ℵ)))
19 iscard3 8772 . . . . . . . . . 10 ((card‘ (𝐹𝐴)) = (𝐹𝐴) ↔ (𝐹𝐴) ∈ (ω ∪ ran ℵ))
2018, 19syl6ibr 240 . . . . . . . . 9 (𝐴 ∈ V → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (card‘ (𝐹𝐴)) = (𝐹𝐴)))
2120adantrd 482 . . . . . . . 8 (𝐴 ∈ V → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ (𝑥𝐴 ∧ (𝐹𝑥) ∈ ran ℵ)) → (card‘ (𝐹𝐴)) = (𝐹𝐴)))
2217, 21jcad 553 . . . . . . 7 (𝐴 ∈ V → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ (𝑥𝐴 ∧ (𝐹𝑥) ∈ ran ℵ)) → (ω ⊆ (𝐹𝐴) ∧ (card‘ (𝐹𝐴)) = (𝐹𝐴))))
23 isinfcard 8771 . . . . . . 7 ((ω ⊆ (𝐹𝐴) ∧ (card‘ (𝐹𝐴)) = (𝐹𝐴)) ↔ (𝐹𝐴) ∈ ran ℵ)
2422, 23syl6ib 239 . . . . . 6 (𝐴 ∈ V → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ (𝑥𝐴 ∧ (𝐹𝑥) ∈ ran ℵ)) → (𝐹𝐴) ∈ ran ℵ))
2524exp4d 634 . . . . 5 (𝐴 ∈ V → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝑥𝐴 → ((𝐹𝑥) ∈ ran ℵ → (𝐹𝐴) ∈ ran ℵ))))
2625imp 443 . . . 4 ((𝐴 ∈ V ∧ 𝐹:𝐴⟶(ω ∪ ran ℵ)) → (𝑥𝐴 → ((𝐹𝑥) ∈ ran ℵ → (𝐹𝐴) ∈ ran ℵ)))
2726rexlimdv 3007 . . 3 ((𝐴 ∈ V ∧ 𝐹:𝐴⟶(ω ∪ ran ℵ)) → (∃𝑥𝐴 (𝐹𝑥) ∈ ran ℵ → (𝐹𝐴) ∈ ran ℵ))
2827expimpd 626 . 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 382   = wceq 1474  wcel 1975  wrex 2892  Vcvv 3168  cun 3533  wss 3535   cuni 4362  ran crn 5025  cima 5027   Fn wfn 5781  wf 5782  cfv 5786  ωcom 6930  cardccrd 8617  cale 8618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-inf2 8394
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-se 4984  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-isom 5795  df-riota 6485  df-om 6931  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-er 7602  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-oi 8271  df-har 8319  df-card 8621  df-aleph 8622
This theorem is referenced by:  alephfplem4  8786
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