Theorem iunmapdisj 8846

Description: The union ∪ 𝑛 ∈ 𝐶(𝐴 ↑_𝑚 𝑛) is a disjoint union. (Contributed by Mario Carneiro, 17-May-2015.) (Revised by NM, 16-Jun-2017.)

Assertion

Ref	Expression
iunmapdisj	⊢ ∃*𝑛 ∈ 𝐶 𝐵 ∈ (𝐴 ↑𝑚 𝑛)

Distinct variable group:   𝐵,𝑛

Allowed substitution hints:   𝐴(𝑛)   𝐶(𝑛)

Proof of Theorem iunmapdisj

Step	Hyp	Ref	Expression
1		moeq 3382	. . . 4 ⊢ ∃*𝑛 𝑛 = dom 𝐵
2		elmapi 7879	. . . . . 6 ⊢ (𝐵 ∈ (𝐴 ↑𝑚 𝑛) → 𝐵:𝑛⟶𝐴)
3		fdm 6051	. . . . . . 7 ⊢ (𝐵:𝑛⟶𝐴 → dom 𝐵 = 𝑛)
4	3	eqcomd 2628	. . . . . 6 ⊢ (𝐵:𝑛⟶𝐴 → 𝑛 = dom 𝐵)
5	2, 4	syl 17	. . . . 5 ⊢ (𝐵 ∈ (𝐴 ↑𝑚 𝑛) → 𝑛 = dom 𝐵)
6	5	moimi 2520	. . . 4 ⊢ (∃*𝑛 𝑛 = dom 𝐵 → ∃*𝑛 𝐵 ∈ (𝐴 ↑𝑚 𝑛))
7	1, 6	ax-mp 5	. . 3 ⊢ ∃*𝑛 𝐵 ∈ (𝐴 ↑𝑚 𝑛)
8	7	moani 2525	. 2 ⊢ ∃*𝑛(𝑛 ∈ 𝐶 ∧ 𝐵 ∈ (𝐴 ↑𝑚 𝑛))
9		df-rmo 2920	. 2 ⊢ (∃*𝑛 ∈ 𝐶 𝐵 ∈ (𝐴 ↑𝑚 𝑛) ↔ ∃*𝑛(𝑛 ∈ 𝐶 ∧ 𝐵 ∈ (𝐴 ↑𝑚 𝑛)))
10	8, 9	mpbir 221	1 ⊢ ∃*𝑛 ∈ 𝐶 𝐵 ∈ (𝐴 ↑𝑚 𝑛)

Syntax hints:   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃*wmo 2471  ∃*wrmo 2915  dom cdm 5114  ⟶wf 5884  (class class class)co 6650   ↑_𝑚 cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859

This theorem is referenced by: (None)