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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 (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 3480 . 2 (𝐴𝐵𝐴 ∈ V)
2 isinfcard 10117 . . . . . . . . . . . . 13 ((ω ⊆ (𝐹𝑥) ∧ (card‘(𝐹𝑥)) = (𝐹𝑥)) ↔ (𝐹𝑥) ∈ ran ℵ)
32bicomi 223 . . . . . . . . . . . 12 ((𝐹𝑥) ∈ ran ℵ ↔ (ω ⊆ (𝐹𝑥) ∧ (card‘(𝐹𝑥)) = (𝐹𝑥)))
43simplbi 496 . . . . . . . . . . 11 ((𝐹𝑥) ∈ ran ℵ → ω ⊆ (𝐹𝑥))
5 ffn 6723 . . . . . . . . . . . 12 (𝐹:𝐴⟶(ω ∪ ran ℵ) → 𝐹 Fn 𝐴)
6 fnfvelrn 7089 . . . . . . . . . . . . . . . 16 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹𝑥) ∈ ran 𝐹)
76ex 411 . . . . . . . . . . . . . . 15 (𝐹 Fn 𝐴 → (𝑥𝐴 → (𝐹𝑥) ∈ ran 𝐹))
8 fnima 6686 . . . . . . . . . . . . . . . 16 (𝐹 Fn 𝐴 → (𝐹𝐴) = ran 𝐹)
98eleq2d 2811 . . . . . . . . . . . . . . 15 (𝐹 Fn 𝐴 → ((𝐹𝑥) ∈ (𝐹𝐴) ↔ (𝐹𝑥) ∈ ran 𝐹))
107, 9sylibrd 258 . . . . . . . . . . . . . 14 (𝐹 Fn 𝐴 → (𝑥𝐴 → (𝐹𝑥) ∈ (𝐹𝐴)))
11 elssuni 4941 . . . . . . . . . . . . . 14 ((𝐹𝑥) ∈ (𝐹𝐴) → (𝐹𝑥) ⊆ (𝐹𝐴))
1210, 11syl6 35 . . . . . . . . . . . . 13 (𝐹 Fn 𝐴 → (𝑥𝐴 → (𝐹𝑥) ⊆ (𝐹𝐴)))
1312imp 405 . . . . . . . . . . . 12 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹𝑥) ⊆ (𝐹𝐴))
145, 13sylan 578 . . . . . . . . . . 11 ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ 𝑥𝐴) → (𝐹𝑥) ⊆ (𝐹𝐴))
154, 14sylan9ssr 3991 . . . . . . . . . 10 (((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ 𝑥𝐴) ∧ (𝐹𝑥) ∈ ran ℵ) → ω ⊆ (𝐹𝐴))
1615anasss 465 . . . . . . . . 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 10121 . . . . . . . . . 10 (𝐴 ∈ V → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹𝐴) ∈ (ω ∪ ran ℵ)))
19 iscard3 10118 . . . . . . . . . 10 ((card‘ (𝐹𝐴)) = (𝐹𝐴) ↔ (𝐹𝐴) ∈ (ω ∪ ran ℵ))
2018, 19imbitrrdi 251 . . . . . . . . 9 (𝐴 ∈ V → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (card‘ (𝐹𝐴)) = (𝐹𝐴)))
2120adantrd 490 . . . . . . . 8 (𝐴 ∈ V → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ (𝑥𝐴 ∧ (𝐹𝑥) ∈ ran ℵ)) → (card‘ (𝐹𝐴)) = (𝐹𝐴)))
2217, 21jcad 511 . . . . . . 7 (𝐴 ∈ V → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ (𝑥𝐴 ∧ (𝐹𝑥) ∈ ran ℵ)) → (ω ⊆ (𝐹𝐴) ∧ (card‘ (𝐹𝐴)) = (𝐹𝐴))))
23 isinfcard 10117 . . . . . . 7 ((ω ⊆ (𝐹𝐴) ∧ (card‘ (𝐹𝐴)) = (𝐹𝐴)) ↔ (𝐹𝐴) ∈ ran ℵ)
2422, 23imbitrdi 250 . . . . . 6 (𝐴 ∈ V → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ (𝑥𝐴 ∧ (𝐹𝑥) ∈ ran ℵ)) → (𝐹𝐴) ∈ ran ℵ))
2524exp4d 432 . . . . 5 (𝐴 ∈ V → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝑥𝐴 → ((𝐹𝑥) ∈ ran ℵ → (𝐹𝐴) ∈ ran ℵ))))
2625imp 405 . . . 4 ((𝐴 ∈ V ∧ 𝐹:𝐴⟶(ω ∪ ran ℵ)) → (𝑥𝐴 → ((𝐹𝑥) ∈ ran ℵ → (𝐹𝐴) ∈ ran ℵ)))
2726rexlimdv 3142 . . 3 ((𝐴 ∈ V ∧ 𝐹:𝐴⟶(ω ∪ ran ℵ)) → (∃𝑥𝐴 (𝐹𝑥) ∈ ran ℵ → (𝐹𝐴) ∈ ran ℵ))
2827expimpd 452 . 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 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  cale 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
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