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Theorem cardinfima 9515
 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 3511 . 2 (𝐴𝐵𝐴 ∈ V)
2 isinfcard 9510 . . . . . . . . . . . . 13 ((ω ⊆ (𝐹𝑥) ∧ (card‘(𝐹𝑥)) = (𝐹𝑥)) ↔ (𝐹𝑥) ∈ ran ℵ)
32bicomi 226 . . . . . . . . . . . 12 ((𝐹𝑥) ∈ ran ℵ ↔ (ω ⊆ (𝐹𝑥) ∧ (card‘(𝐹𝑥)) = (𝐹𝑥)))
43simplbi 500 . . . . . . . . . . 11 ((𝐹𝑥) ∈ ran ℵ → ω ⊆ (𝐹𝑥))
5 ffn 6507 . . . . . . . . . . . 12 (𝐹:𝐴⟶(ω ∪ ran ℵ) → 𝐹 Fn 𝐴)
6 fnfvelrn 6841 . . . . . . . . . . . . . . . 16 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹𝑥) ∈ ran 𝐹)
76ex 415 . . . . . . . . . . . . . . 15 (𝐹 Fn 𝐴 → (𝑥𝐴 → (𝐹𝑥) ∈ ran 𝐹))
8 fnima 6471 . . . . . . . . . . . . . . . 16 (𝐹 Fn 𝐴 → (𝐹𝐴) = ran 𝐹)
98eleq2d 2896 . . . . . . . . . . . . . . 15 (𝐹 Fn 𝐴 → ((𝐹𝑥) ∈ (𝐹𝐴) ↔ (𝐹𝑥) ∈ ran 𝐹))
107, 9sylibrd 261 . . . . . . . . . . . . . 14 (𝐹 Fn 𝐴 → (𝑥𝐴 → (𝐹𝑥) ∈ (𝐹𝐴)))
11 elssuni 4859 . . . . . . . . . . . . . 14 ((𝐹𝑥) ∈ (𝐹𝐴) → (𝐹𝑥) ⊆ (𝐹𝐴))
1210, 11syl6 35 . . . . . . . . . . . . 13 (𝐹 Fn 𝐴 → (𝑥𝐴 → (𝐹𝑥) ⊆ (𝐹𝐴)))
1312imp 409 . . . . . . . . . . . 12 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹𝑥) ⊆ (𝐹𝐴))
145, 13sylan 582 . . . . . . . . . . 11 ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ 𝑥𝐴) → (𝐹𝑥) ⊆ (𝐹𝐴))
154, 14sylan9ssr 3979 . . . . . . . . . 10 (((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ 𝑥𝐴) ∧ (𝐹𝑥) ∈ ran ℵ) → ω ⊆ (𝐹𝐴))
1615anasss 469 . . . . . . . . 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 9514 . . . . . . . . . 10 (𝐴 ∈ V → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹𝐴) ∈ (ω ∪ ran ℵ)))
19 iscard3 9511 . . . . . . . . . 10 ((card‘ (𝐹𝐴)) = (𝐹𝐴) ↔ (𝐹𝐴) ∈ (ω ∪ ran ℵ))
2018, 19syl6ibr 254 . . . . . . . . 9 (𝐴 ∈ V → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (card‘ (𝐹𝐴)) = (𝐹𝐴)))
2120adantrd 494 . . . . . . . 8 (𝐴 ∈ V → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ (𝑥𝐴 ∧ (𝐹𝑥) ∈ ran ℵ)) → (card‘ (𝐹𝐴)) = (𝐹𝐴)))
2217, 21jcad 515 . . . . . . 7 (𝐴 ∈ V → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ (𝑥𝐴 ∧ (𝐹𝑥) ∈ ran ℵ)) → (ω ⊆ (𝐹𝐴) ∧ (card‘ (𝐹𝐴)) = (𝐹𝐴))))
23 isinfcard 9510 . . . . . . 7 ((ω ⊆ (𝐹𝐴) ∧ (card‘ (𝐹𝐴)) = (𝐹𝐴)) ↔ (𝐹𝐴) ∈ ran ℵ)
2422, 23syl6ib 253 . . . . . 6 (𝐴 ∈ V → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ (𝑥𝐴 ∧ (𝐹𝑥) ∈ ran ℵ)) → (𝐹𝐴) ∈ ran ℵ))
2524exp4d 436 . . . . 5 (𝐴 ∈ V → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝑥𝐴 → ((𝐹𝑥) ∈ ran ℵ → (𝐹𝐴) ∈ ran ℵ))))
2625imp 409 . . . 4 ((𝐴 ∈ V ∧ 𝐹:𝐴⟶(ω ∪ ran ℵ)) → (𝑥𝐴 → ((𝐹𝑥) ∈ ran ℵ → (𝐹𝐴) ∈ ran ℵ)))
2726rexlimdv 3281 . . 3 ((𝐴 ∈ V ∧ 𝐹:𝐴⟶(ω ∪ ran ℵ)) → (∃𝑥𝐴 (𝐹𝑥) ∈ ran ℵ → (𝐹𝐴) ∈ ran ℵ))
2827expimpd 456 . 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 398   = wceq 1530   ∈ wcel 2107  ∃wrex 3137  Vcvv 3493   ∪ cun 3932   ⊆ wss 3934  ∪ cuni 4830  ran crn 5549   “ cima 5551   Fn wfn 6343  ⟶wf 6344  ‘cfv 6348  ωcom 7572  cardccrd 9356  ℵcale 9357 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-inf2 9096 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-om 7573  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-oi 8966  df-har 9014  df-card 9360  df-aleph 9361 This theorem is referenced by:  alephfplem4  9525
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