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Theorem iunmapdisj 10061
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 3716 . . . 4 ∃*𝑛 𝑛 = dom 𝐵
2 elmapi 8888 . . . . . 6 (𝐵 ∈ (𝐴m 𝑛) → 𝐵:𝑛𝐴)
3 fdm 6746 . . . . . . 7 (𝐵:𝑛𝐴 → dom 𝐵 = 𝑛)
43eqcomd 2741 . . . . . 6 (𝐵:𝑛𝐴𝑛 = dom 𝐵)
52, 4syl 17 . . . . 5 (𝐵 ∈ (𝐴m 𝑛) → 𝑛 = dom 𝐵)
65moimi 2543 . . . 4 (∃*𝑛 𝑛 = dom 𝐵 → ∃*𝑛 𝐵 ∈ (𝐴m 𝑛))
71, 6ax-mp 5 . . 3 ∃*𝑛 𝐵 ∈ (𝐴m 𝑛)
87moani 2551 . 2 ∃*𝑛(𝑛𝐶𝐵 ∈ (𝐴m 𝑛))
9 df-rmo 3378 . 2 (∃*𝑛𝐶 𝐵 ∈ (𝐴m 𝑛) ↔ ∃*𝑛(𝑛𝐶𝐵 ∈ (𝐴m 𝑛)))
108, 9mpbir 231 1 ∃*𝑛𝐶 𝐵 ∈ (𝐴m 𝑛)
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wcel 2106  ∃*wmo 2536  ∃*wrmo 3377  dom cdm 5689  wf 6559  (class class class)co 7431  m cmap 8865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rmo 3378  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867
This theorem is referenced by: (None)
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