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Theorem cardinfima 10092
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 ∈ 𝐡 β†’ ((𝐹:𝐴⟢(Ο‰ βˆͺ ran β„΅) ∧ βˆƒπ‘₯ ∈ 𝐴 (πΉβ€˜π‘₯) ∈ ran β„΅) β†’ βˆͺ (𝐹 β€œ 𝐴) ∈ ran β„΅))
Distinct variable groups:   π‘₯,𝐹   π‘₯,𝐴
Allowed substitution hint:   𝐡(π‘₯)
Proof of Theorem cardinfima
StepHypRef Expression
1 elex 3493 . 2 (𝐴 ∈ 𝐡 β†’ 𝐴 ∈ V)
2 isinfcard 10087 . . . . . . . . . . . . 13 ((Ο‰ βŠ† (πΉβ€˜π‘₯) ∧ (cardβ€˜(πΉβ€˜π‘₯)) = (πΉβ€˜π‘₯)) ↔ (πΉβ€˜π‘₯) ∈ ran β„΅)
32bicomi 223 . . . . . . . . . . . 12 ((πΉβ€˜π‘₯) ∈ ran β„΅ ↔ (Ο‰ βŠ† (πΉβ€˜π‘₯) ∧ (cardβ€˜(πΉβ€˜π‘₯)) = (πΉβ€˜π‘₯)))
43simplbi 499 . . . . . . . . . . 11 ((πΉβ€˜π‘₯) ∈ ran β„΅ β†’ Ο‰ βŠ† (πΉβ€˜π‘₯))
5 ffn 6718 . . . . . . . . . . . 12 (𝐹:𝐴⟢(Ο‰ βˆͺ ran β„΅) β†’ 𝐹 Fn 𝐴)
6 fnfvelrn 7083 . . . . . . . . . . . . . . . 16 ((𝐹 Fn 𝐴 ∧ π‘₯ ∈ 𝐴) β†’ (πΉβ€˜π‘₯) ∈ ran 𝐹)
76ex 414 . . . . . . . . . . . . . . 15 (𝐹 Fn 𝐴 β†’ (π‘₯ ∈ 𝐴 β†’ (πΉβ€˜π‘₯) ∈ ran 𝐹))
8 fnima 6681 . . . . . . . . . . . . . . . 16 (𝐹 Fn 𝐴 β†’ (𝐹 β€œ 𝐴) = ran 𝐹)
98eleq2d 2820 . . . . . . . . . . . . . . 15 (𝐹 Fn 𝐴 β†’ ((πΉβ€˜π‘₯) ∈ (𝐹 β€œ 𝐴) ↔ (πΉβ€˜π‘₯) ∈ ran 𝐹))
107, 9sylibrd 259 . . . . . . . . . . . . . 14 (𝐹 Fn 𝐴 β†’ (π‘₯ ∈ 𝐴 β†’ (πΉβ€˜π‘₯) ∈ (𝐹 β€œ 𝐴)))
11 elssuni 4942 . . . . . . . . . . . . . 14 ((πΉβ€˜π‘₯) ∈ (𝐹 β€œ 𝐴) β†’ (πΉβ€˜π‘₯) βŠ† βˆͺ (𝐹 β€œ 𝐴))
1210, 11syl6 35 . . . . . . . . . . . . 13 (𝐹 Fn 𝐴 β†’ (π‘₯ ∈ 𝐴 β†’ (πΉβ€˜π‘₯) βŠ† βˆͺ (𝐹 β€œ 𝐴)))
1312imp 408 . . . . . . . . . . . 12 ((𝐹 Fn 𝐴 ∧ π‘₯ ∈ 𝐴) β†’ (πΉβ€˜π‘₯) βŠ† βˆͺ (𝐹 β€œ 𝐴))
145, 13sylan 581 . . . . . . . . . . 11 ((𝐹:𝐴⟢(Ο‰ βˆͺ ran β„΅) ∧ π‘₯ ∈ 𝐴) β†’ (πΉβ€˜π‘₯) βŠ† βˆͺ (𝐹 β€œ 𝐴))
154, 14sylan9ssr 3997 . . . . . . . . . 10 (((𝐹:𝐴⟢(Ο‰ βˆͺ ran β„΅) ∧ π‘₯ ∈ 𝐴) ∧ (πΉβ€˜π‘₯) ∈ ran β„΅) β†’ Ο‰ βŠ† βˆͺ (𝐹 β€œ 𝐴))
1615anasss 468 . . . . . . . . 9 ((𝐹:𝐴⟢(Ο‰ βˆͺ ran β„΅) ∧ (π‘₯ ∈ 𝐴 ∧ (πΉβ€˜π‘₯) ∈ ran β„΅)) β†’ Ο‰ βŠ† βˆͺ (𝐹 β€œ 𝐴))
1716a1i 11 . . . . . . . 8 (𝐴 ∈ V β†’ ((𝐹:𝐴⟢(Ο‰ βˆͺ ran β„΅) ∧ (π‘₯ ∈ 𝐴 ∧ (πΉβ€˜π‘₯) ∈ ran β„΅)) β†’ Ο‰ βŠ† βˆͺ (𝐹 β€œ 𝐴)))
18 carduniima 10091 . . . . . . . . . 10 (𝐴 ∈ V β†’ (𝐹:𝐴⟢(Ο‰ βˆͺ ran β„΅) β†’ βˆͺ (𝐹 β€œ 𝐴) ∈ (Ο‰ βˆͺ ran β„΅)))
19 iscard3 10088 . . . . . . . . . 10 ((cardβ€˜βˆͺ (𝐹 β€œ 𝐴)) = βˆͺ (𝐹 β€œ 𝐴) ↔ βˆͺ (𝐹 β€œ 𝐴) ∈ (Ο‰ βˆͺ ran β„΅))
2018, 19imbitrrdi 251 . . . . . . . . 9 (𝐴 ∈ V β†’ (𝐹:𝐴⟢(Ο‰ βˆͺ ran β„΅) β†’ (cardβ€˜βˆͺ (𝐹 β€œ 𝐴)) = βˆͺ (𝐹 β€œ 𝐴)))
2120adantrd 493 . . . . . . . 8 (𝐴 ∈ V β†’ ((𝐹:𝐴⟢(Ο‰ βˆͺ ran β„΅) ∧ (π‘₯ ∈ 𝐴 ∧ (πΉβ€˜π‘₯) ∈ ran β„΅)) β†’ (cardβ€˜βˆͺ (𝐹 β€œ 𝐴)) = βˆͺ (𝐹 β€œ 𝐴)))
2217, 21jcad 514 . . . . . . 7 (𝐴 ∈ V β†’ ((𝐹:𝐴⟢(Ο‰ βˆͺ ran β„΅) ∧ (π‘₯ ∈ 𝐴 ∧ (πΉβ€˜π‘₯) ∈ ran β„΅)) β†’ (Ο‰ βŠ† βˆͺ (𝐹 β€œ 𝐴) ∧ (cardβ€˜βˆͺ (𝐹 β€œ 𝐴)) = βˆͺ (𝐹 β€œ 𝐴))))
23 isinfcard 10087 . . . . . . 7 ((Ο‰ βŠ† βˆͺ (𝐹 β€œ 𝐴) ∧ (cardβ€˜βˆͺ (𝐹 β€œ 𝐴)) = βˆͺ (𝐹 β€œ 𝐴)) ↔ βˆͺ (𝐹 β€œ 𝐴) ∈ ran β„΅)
2422, 23imbitrdi 250 . . . . . 6 (𝐴 ∈ V β†’ ((𝐹:𝐴⟢(Ο‰ βˆͺ ran β„΅) ∧ (π‘₯ ∈ 𝐴 ∧ (πΉβ€˜π‘₯) ∈ ran β„΅)) β†’ βˆͺ (𝐹 β€œ 𝐴) ∈ ran β„΅))
2524exp4d 435 . . . . 5 (𝐴 ∈ V β†’ (𝐹:𝐴⟢(Ο‰ βˆͺ ran β„΅) β†’ (π‘₯ ∈ 𝐴 β†’ ((πΉβ€˜π‘₯) ∈ ran β„΅ β†’ βˆͺ (𝐹 β€œ 𝐴) ∈ ran β„΅))))
2625imp 408 . . . 4 ((𝐴 ∈ V ∧ 𝐹:𝐴⟢(Ο‰ βˆͺ ran β„΅)) β†’ (π‘₯ ∈ 𝐴 β†’ ((πΉβ€˜π‘₯) ∈ ran β„΅ β†’ βˆͺ (𝐹 β€œ 𝐴) ∈ ran β„΅)))
2726rexlimdv 3154 . . 3 ((𝐴 ∈ V ∧ 𝐹:𝐴⟢(Ο‰ βˆͺ ran β„΅)) β†’ (βˆƒπ‘₯ ∈ 𝐴 (πΉβ€˜π‘₯) ∈ ran β„΅ β†’ βˆͺ (𝐹 β€œ 𝐴) ∈ ran β„΅))
2827expimpd 455 . 2 (𝐴 ∈ V β†’ ((𝐹:𝐴⟢(Ο‰ βˆͺ ran β„΅) ∧ βˆƒπ‘₯ ∈ 𝐴 (πΉβ€˜π‘₯) ∈ ran β„΅) β†’ βˆͺ (𝐹 β€œ 𝐴) ∈ ran β„΅))
291, 28syl 17 1 (𝐴 ∈ 𝐡 β†’ ((𝐹:𝐴⟢(Ο‰ βˆͺ ran β„΅) ∧ βˆƒπ‘₯ ∈ 𝐴 (πΉβ€˜π‘₯) ∈ ran β„΅) β†’ βˆͺ (𝐹 β€œ 𝐴) ∈ ran β„΅))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071  Vcvv 3475   βˆͺ cun 3947   βŠ† wss 3949  βˆͺ cuni 4909  ran crn 5678   β€œ cima 5680   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  Ο‰com 7855  cardccrd 9930  β„΅cale 9931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-oi 9505  df-har 9552  df-card 9934  df-aleph 9935
This theorem is referenced by:  alephfplem4  10102
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