Theorem carduniima 3695

Description: The union of the image of a mapping to cardinals is a cardinal. Proposition 11.16 of [TakeutiZaring] p. 104.

Assertion

Ref	Expression
carduniima	⊢ (A ∈ B → (F:A–→(ω ∪ ran ℵ) → ∪(F “ A) ∈ (ω ∪ ran ℵ)))

Proof of Theorem carduniima

Step	Hyp	Ref	Expression
1		funimaexg 2715	. . . 4 ⊢ (A ∈ B → (Fun F → (F “ A) ∈ V))
2		ffun 2754	. . . 4 ⊢ (F:A–→(ω ∪ ran ℵ) → Fun F)
3	1, 2	syl5 22	. . 3 ⊢ (A ∈ B → (F:A–→(ω ∪ ran ℵ) → (F “ A) ∈ V))
4		carduni 3664	. . . 4 ⊢ ((F “ A) ∈ V → (∀x ∈ (F “ A)(card ‘x) = x → (card ‘∪(F “ A)) = ∪(F “ A)))
5		ffn 2752	. . . . . . . . 9 ⊢ (F:A–→(ω ∪ ran ℵ) → F Fn A)
6		fnima 2738	. . . . . . . . 9 ⊢ (F Fn A → (F “ A) = ran F)
7	5, 6	syl 12	. . . . . . . 8 ⊢ (F:A–→(ω ∪ ran ℵ) → (F “ A) = ran F)
8		frn 2757	. . . . . . . 8 ⊢ (F:A–→(ω ∪ ran ℵ) → ran F ⊆ (ω ∪ ran ℵ))
9	7, 8	eqsstrd 1534	. . . . . . 7 ⊢ (F:A–→(ω ∪ ran ℵ) → (F “ A) ⊆ (ω ∪ ran ℵ))
10	9	sseld 1506	. . . . . 6 ⊢ (F:A–→(ω ∪ ran ℵ) → (x ∈ (F “ A) → x ∈ (ω ∪ ran ℵ)))
11		iscard3 3693	. . . . . 6 ⊢ ((card ‘x) = x ↔ x ∈ (ω ∪ ran ℵ))
12	10, 11	syl6ibr 186	. . . . 5 ⊢ (F:A–→(ω ∪ ran ℵ) → (x ∈ (F “ A) → (card ‘x) = x))
13	12	r19.21aiv 1259	. . . 4 ⊢ (F:A–→(ω ∪ ran ℵ) → ∀x ∈ (F “ A)(card ‘x) = x)
14	4, 13	syl5 22	. . 3 ⊢ ((F “ A) ∈ V → (F:A–→(ω ∪ ran ℵ) → (card ‘∪(F “ A)) = ∪(F “ A)))
15	3, 14	syli 52	. 2 ⊢ (A ∈ B → (F:A–→(ω ∪ ran ℵ) → (card ‘∪(F “ A)) = ∪(F “ A)))
16		iscard3 3693	. 2 ⊢ ((card ‘∪(F “ A)) = ∪(F “ A) ↔ ∪(F “ A) ∈ (ω ∪ ran ℵ))
17	15, 16	syl6ib 185	1 ⊢ (A ∈ B → (F:A–→(ω ∪ ran ℵ) → ∪(F “ A) ∈ (ω ∪ ran ℵ)))

Syntax hints:   → wi 2   = wceq 1091   ∈ wcel 1092  ∀wral 1201  Vcvv 1348   ∪ cun 1485  ∪cuni 1919  ωcom 2372  ran crn 2411   “ cima 2413  Fun wfun 2416   Fn wfn 2417  –→wf 2418   ‘cfv 2422  cardccrd 3620  ℵcale 3621

This theorem is referenced by:  cardinfima 3696

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-er 3200  df-en 3274  df-dom 3275  df-sdom 3276  df-card 3623  df-aleph 3624

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