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Theorem iunmapdisj 7650
Description: The union  U_ n  e.  C ( A  ^m  n ) is a disjoint union. (Contributed by Mario Carneiro, 17-May-2015.) (Revised by NM, 16-Jun-2017.)
Assertion
Ref Expression
iunmapdisj  |-  E* n  e.  C B  e.  ( A  ^m  n )
Distinct variable group:    B, n
Allowed substitution hints:    A( n)    C( n)

Proof of Theorem iunmapdisj
StepHypRef Expression
1 moeq 2941 . . . 4  |-  E* n  n  =  dom  B
2 elmapi 6792 . . . . . 6  |-  ( B  e.  ( A  ^m  n )  ->  B : n --> A )
3 fdm 5393 . . . . . . 7  |-  ( B : n --> A  ->  dom  B  =  n )
43eqcomd 2288 . . . . . 6  |-  ( B : n --> A  ->  n  =  dom  B )
52, 4syl 15 . . . . 5  |-  ( B  e.  ( A  ^m  n )  ->  n  =  dom  B )
65moimi 2190 . . . 4  |-  ( E* n  n  =  dom  B  ->  E* n  B  e.  ( A  ^m  n ) )
71, 6ax-mp 8 . . 3  |-  E* n  B  e.  ( A  ^m  n )
87moani 2195 . 2  |-  E* n
( n  e.  C  /\  B  e.  ( A  ^m  n ) )
9 df-rmo 2551 . 2  |-  ( E* n  e.  C B  e.  ( A  ^m  n )  <->  E* n
( n  e.  C  /\  B  e.  ( A  ^m  n ) ) )
108, 9mpbir 200 1  |-  E* n  e.  C B  e.  ( A  ^m  n )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684   E*wmo 2144   E*wrmo 2546   dom cdm 4689   -->wf 5251  (class class class)co 5858    ^m cmap 6772
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774
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