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Theorem iunmapdisj 9007
Description: The union 𝑛𝐶(𝐴𝑚 𝑛) is a disjoint union. (Contributed by Mario Carneiro, 17-May-2015.) (Revised by NM, 16-Jun-2017.)
Assertion
Ref Expression
iunmapdisj ∃*𝑛𝐶 𝐵 ∈ (𝐴𝑚 𝑛)
Distinct variable group:   𝐵,𝑛
Allowed substitution hints:   𝐴(𝑛)   𝐶(𝑛)

Proof of Theorem iunmapdisj
StepHypRef Expression
1 moeq 3511 . . . 4 ∃*𝑛 𝑛 = dom 𝐵
2 elmapi 8033 . . . . . 6 (𝐵 ∈ (𝐴𝑚 𝑛) → 𝐵:𝑛𝐴)
3 fdm 6200 . . . . . . 7 (𝐵:𝑛𝐴 → dom 𝐵 = 𝑛)
43eqcomd 2754 . . . . . 6 (𝐵:𝑛𝐴𝑛 = dom 𝐵)
52, 4syl 17 . . . . 5 (𝐵 ∈ (𝐴𝑚 𝑛) → 𝑛 = dom 𝐵)
65moimi 2646 . . . 4 (∃*𝑛 𝑛 = dom 𝐵 → ∃*𝑛 𝐵 ∈ (𝐴𝑚 𝑛))
71, 6ax-mp 5 . . 3 ∃*𝑛 𝐵 ∈ (𝐴𝑚 𝑛)
87moani 2651 . 2 ∃*𝑛(𝑛𝐶𝐵 ∈ (𝐴𝑚 𝑛))
9 df-rmo 3046 . 2 (∃*𝑛𝐶 𝐵 ∈ (𝐴𝑚 𝑛) ↔ ∃*𝑛(𝑛𝐶𝐵 ∈ (𝐴𝑚 𝑛)))
108, 9mpbir 221 1 ∃*𝑛𝐶 𝐵 ∈ (𝐴𝑚 𝑛)
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1620  wcel 2127  ∃*wmo 2596  ∃*wrmo 3041  dom cdm 5254  wf 6033  (class class class)co 6801  𝑚 cmap 8011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-rmo 3046  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-fv 6045  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-1st 7321  df-2nd 7322  df-map 8013
This theorem is referenced by: (None)
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