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Theorem cardinfima 10094
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 3491 . 2 (𝐴 ∈ 𝐡 β†’ 𝐴 ∈ V)
2 isinfcard 10089 . . . . . . . . . . . . 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 496 . . . . . . . . . . 11 ((πΉβ€˜π‘₯) ∈ ran β„΅ β†’ Ο‰ βŠ† (πΉβ€˜π‘₯))
5 ffn 6716 . . . . . . . . . . . 12 (𝐹:𝐴⟢(Ο‰ βˆͺ ran β„΅) β†’ 𝐹 Fn 𝐴)
6 fnfvelrn 7081 . . . . . . . . . . . . . . . 16 ((𝐹 Fn 𝐴 ∧ π‘₯ ∈ 𝐴) β†’ (πΉβ€˜π‘₯) ∈ ran 𝐹)
76ex 411 . . . . . . . . . . . . . . 15 (𝐹 Fn 𝐴 β†’ (π‘₯ ∈ 𝐴 β†’ (πΉβ€˜π‘₯) ∈ ran 𝐹))
8 fnima 6679 . . . . . . . . . . . . . . . 16 (𝐹 Fn 𝐴 β†’ (𝐹 β€œ 𝐴) = ran 𝐹)
98eleq2d 2817 . . . . . . . . . . . . . . 15 (𝐹 Fn 𝐴 β†’ ((πΉβ€˜π‘₯) ∈ (𝐹 β€œ 𝐴) ↔ (πΉβ€˜π‘₯) ∈ ran 𝐹))
107, 9sylibrd 258 . . . . . . . . . . . . . 14 (𝐹 Fn 𝐴 β†’ (π‘₯ ∈ 𝐴 β†’ (πΉβ€˜π‘₯) ∈ (𝐹 β€œ 𝐴)))
11 elssuni 4940 . . . . . . . . . . . . . 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 405 . . . . . . . . . . . 12 ((𝐹 Fn 𝐴 ∧ π‘₯ ∈ 𝐴) β†’ (πΉβ€˜π‘₯) βŠ† βˆͺ (𝐹 β€œ 𝐴))
145, 13sylan 578 . . . . . . . . . . 11 ((𝐹:𝐴⟢(Ο‰ βˆͺ ran β„΅) ∧ π‘₯ ∈ 𝐴) β†’ (πΉβ€˜π‘₯) βŠ† βˆͺ (𝐹 β€œ 𝐴))
154, 14sylan9ssr 3995 . . . . . . . . . 10 (((𝐹:𝐴⟢(Ο‰ βˆͺ ran β„΅) ∧ π‘₯ ∈ 𝐴) ∧ (πΉβ€˜π‘₯) ∈ ran β„΅) β†’ Ο‰ βŠ† βˆͺ (𝐹 β€œ 𝐴))
1615anasss 465 . . . . . . . . 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 10093 . . . . . . . . . 10 (𝐴 ∈ V β†’ (𝐹:𝐴⟢(Ο‰ βˆͺ ran β„΅) β†’ βˆͺ (𝐹 β€œ 𝐴) ∈ (Ο‰ βˆͺ ran β„΅)))
19 iscard3 10090 . . . . . . . . . 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 490 . . . . . . . 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 511 . . . . . . 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 10089 . . . . . . 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 432 . . . . 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 405 . . . 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 3151 . . 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 452 . 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 394   = wceq 1539   ∈ wcel 2104  βˆƒwrex 3068  Vcvv 3472   βˆͺ cun 3945   βŠ† wss 3947  βˆͺ cuni 4907  ran crn 5676   β€œ cima 5678   Fn wfn 6537  βŸΆwf 6538  β€˜cfv 6542  Ο‰com 7857  cardccrd 9932  β„΅cale 9933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-oi 9507  df-har 9554  df-card 9936  df-aleph 9937
This theorem is referenced by:  alephfplem4  10104
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