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Theorem iunmapdisj 9298
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 3635 . . . 4 ∃*𝑛 𝑛 = dom 𝐵
2 elmapi 8281 . . . . . 6 (𝐵 ∈ (𝐴𝑚 𝑛) → 𝐵:𝑛𝐴)
3 fdm 6393 . . . . . . 7 (𝐵:𝑛𝐴 → dom 𝐵 = 𝑛)
43eqcomd 2800 . . . . . 6 (𝐵:𝑛𝐴𝑛 = dom 𝐵)
52, 4syl 17 . . . . 5 (𝐵 ∈ (𝐴𝑚 𝑛) → 𝑛 = dom 𝐵)
65moimi 2580 . . . 4 (∃*𝑛 𝑛 = dom 𝐵 → ∃*𝑛 𝐵 ∈ (𝐴𝑚 𝑛))
71, 6ax-mp 5 . . 3 ∃*𝑛 𝐵 ∈ (𝐴𝑚 𝑛)
87moani 2592 . 2 ∃*𝑛(𝑛𝐶𝐵 ∈ (𝐴𝑚 𝑛))
9 df-rmo 3112 . 2 (∃*𝑛𝐶 𝐵 ∈ (𝐴𝑚 𝑛) ↔ ∃*𝑛(𝑛𝐶𝐵 ∈ (𝐴𝑚 𝑛)))
108, 9mpbir 232 1 ∃*𝑛𝐶 𝐵 ∈ (𝐴𝑚 𝑛)
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1522  wcel 2080  ∃*wmo 2573  ∃*wrmo 3107  dom cdm 5446  wf 6224  (class class class)co 7019  𝑚 cmap 8259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-ral 3109  df-rex 3110  df-rmo 3112  df-rab 3113  df-v 3438  df-sbc 3708  df-csb 3814  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-nul 4214  df-if 4384  df-pw 4457  df-sn 4475  df-pr 4477  df-op 4481  df-uni 4748  df-iun 4829  df-br 4965  df-opab 5027  df-mpt 5044  df-id 5351  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-ima 5459  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-fv 6236  df-ov 7022  df-oprab 7023  df-mpo 7024  df-1st 7548  df-2nd 7549  df-map 8261
This theorem is referenced by: (None)
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