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Theorem iunmapdisj 9438
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 3649 . . . 4 ∃*𝑛 𝑛 = dom 𝐵
2 elmapi 8415 . . . . . 6 (𝐵 ∈ (𝐴m 𝑛) → 𝐵:𝑛𝐴)
3 fdm 6499 . . . . . . 7 (𝐵:𝑛𝐴 → dom 𝐵 = 𝑛)
43eqcomd 2807 . . . . . 6 (𝐵:𝑛𝐴𝑛 = dom 𝐵)
52, 4syl 17 . . . . 5 (𝐵 ∈ (𝐴m 𝑛) → 𝑛 = dom 𝐵)
65moimi 2606 . . . 4 (∃*𝑛 𝑛 = dom 𝐵 → ∃*𝑛 𝐵 ∈ (𝐴m 𝑛))
71, 6ax-mp 5 . . 3 ∃*𝑛 𝐵 ∈ (𝐴m 𝑛)
87moani 2615 . 2 ∃*𝑛(𝑛𝐶𝐵 ∈ (𝐴m 𝑛))
9 df-rmo 3117 . 2 (∃*𝑛𝐶 𝐵 ∈ (𝐴m 𝑛) ↔ ∃*𝑛(𝑛𝐶𝐵 ∈ (𝐴m 𝑛)))
108, 9mpbir 234 1 ∃*𝑛𝐶 𝐵 ∈ (𝐴m 𝑛)
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wcel 2112  ∃*wmo 2599  ∃*wrmo 3112  dom cdm 5523  wf 6324  (class class class)co 7139  m cmap 8393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-map 8395
This theorem is referenced by: (None)
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