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Theorem iunmapdisj 7906
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 3112 . . . 4  |-  E* n  n  =  dom  B
2 elmapi 7040 . . . . . 6  |-  ( B  e.  ( A  ^m  n )  ->  B : n --> A )
3 fdm 5597 . . . . . . 7  |-  ( B : n --> A  ->  dom  B  =  n )
43eqcomd 2443 . . . . . 6  |-  ( B : n --> A  ->  n  =  dom  B )
52, 4syl 16 . . . . 5  |-  ( B  e.  ( A  ^m  n )  ->  n  =  dom  B )
65moimi 2330 . . . 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 2335 . 2  |-  E* n
( n  e.  C  /\  B  e.  ( A  ^m  n ) )
9 df-rmo 2715 . 2  |-  ( E* n  e.  C B  e.  ( A  ^m  n )  <->  E* n
( n  e.  C  /\  B  e.  ( A  ^m  n ) ) )
108, 9mpbir 202 1  |-  E* n  e.  C B  e.  ( A  ^m  n )
Colors of variables: wff set class
Syntax hints:    /\ wa 360    = wceq 1653    e. wcel 1726   E*wmo 2284   E*wrmo 2710   dom cdm 4880   -->wf 5452  (class class class)co 6083    ^m cmap 7020
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 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-map 7022
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