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Theorem iundomg 8417
 Description: An upper bound for the cardinality of an indexed union, with explicit choice principles. depends on and should be thought of as . (Contributed by Mario Carneiro, 1-Sep-2015.)
Hypotheses
Ref Expression
iunfo.1
iundomg.2 AC
iundomg.3
iundomg.4 AC
Assertion
Ref Expression
iundomg
Distinct variable groups:   ,   ,
Allowed substitution hints:   ()   ()   ()

Proof of Theorem iundomg
StepHypRef Expression
1 iunfo.1 . . . . 5
2 iundomg.2 . . . . 5 AC
3 iundomg.3 . . . . 5
41, 2, 3iundom2g 8416 . . . 4
5 iundomg.4 . . . 4 AC
6 acndom2 7936 . . . 4 AC AC
74, 5, 6sylc 59 . . 3 AC
81iunfo 8415 . . 3
9 fodomacn 7938 . . 3 AC
107, 8, 9ee10 1386 . 2
11 domtr 7161 . 2
1210, 4, 11syl2anc 644 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1653   wcel 1726  wral 2706  csn 3815  ciun 4094   class class class wbr 4213   cxp 4877   cres 4881  wfo 5453  (class class class)co 6082  c2nd 6349   cmap 7019   cdom 7108  AC wacn 7826 This theorem is referenced by:  iundom  8418  iunctb  8450 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-map 7021  df-dom 7112  df-acn 7830
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