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Theorem iunmapdisj 9126
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 3577 . . . 4 ∃*𝑛 𝑛 = dom 𝐵
2 elmapi 8111 . . . . . 6 (𝐵 ∈ (𝐴𝑚 𝑛) → 𝐵:𝑛𝐴)
3 fdm 6261 . . . . . . 7 (𝐵:𝑛𝐴 → dom 𝐵 = 𝑛)
43eqcomd 2811 . . . . . 6 (𝐵:𝑛𝐴𝑛 = dom 𝐵)
52, 4syl 17 . . . . 5 (𝐵 ∈ (𝐴𝑚 𝑛) → 𝑛 = dom 𝐵)
65moimi 2683 . . . 4 (∃*𝑛 𝑛 = dom 𝐵 → ∃*𝑛 𝐵 ∈ (𝐴𝑚 𝑛))
71, 6ax-mp 5 . . 3 ∃*𝑛 𝐵 ∈ (𝐴𝑚 𝑛)
87moani 2689 . 2 ∃*𝑛(𝑛𝐶𝐵 ∈ (𝐴𝑚 𝑛))
9 df-rmo 3103 . 2 (∃*𝑛𝐶 𝐵 ∈ (𝐴𝑚 𝑛) ↔ ∃*𝑛(𝑛𝐶𝐵 ∈ (𝐴𝑚 𝑛)))
108, 9mpbir 222 1 ∃*𝑛𝐶 𝐵 ∈ (𝐴𝑚 𝑛)
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1637  wcel 2158  ∃*wmo 2633  ∃*wrmo 3098  dom cdm 5308  wf 6094  (class class class)co 6871  𝑚 cmap 8089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-8 2160  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-sep 4971  ax-nul 4980  ax-pow 5032  ax-pr 5093  ax-un 7176
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-ral 3100  df-rex 3101  df-rmo 3103  df-rab 3104  df-v 3392  df-sbc 3631  df-csb 3726  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-nul 4114  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-op 4374  df-uni 4627  df-iun 4710  df-br 4841  df-opab 4903  df-mpt 4920  df-id 5216  df-xp 5314  df-rel 5315  df-cnv 5316  df-co 5317  df-dm 5318  df-rn 5319  df-res 5320  df-ima 5321  df-iota 6061  df-fun 6100  df-fn 6101  df-f 6102  df-fv 6106  df-ov 6874  df-oprab 6875  df-mpt2 6876  df-1st 7395  df-2nd 7396  df-map 8091
This theorem is referenced by: (None)
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