iunmapdisj - Metamath Proof Explorer

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Theorem iunmapdisj 9917

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 3667	. . . 4 ⊢ ∃*𝑛 𝑛 = dom 𝐵
2		elmapi 8776	. . . . . 6 ⊢ (𝐵 ∈ (𝐴 ↑_m 𝑛) → 𝐵:𝑛⟶𝐴)
3		fdm 6661	. . . . . . 7 ⊢ (𝐵:𝑛⟶𝐴 → dom 𝐵 = 𝑛)
4	3	eqcomd 2735	. . . . . 6 ⊢ (𝐵:𝑛⟶𝐴 → 𝑛 = dom 𝐵)
5	2, 4	syl 17	. . . . 5 ⊢ (𝐵 ∈ (𝐴 ↑_m 𝑛) → 𝑛 = dom 𝐵)
6	5	moimi 2538	. . . 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 2546	. 2 ⊢ ∃*𝑛(𝑛 ∈ 𝐶 ∧ 𝐵 ∈ (𝐴 ↑_m 𝑛))
9		df-rmo 3343	. 2 ⊢ (∃𝑛 ∈ 𝐶 𝐵 ∈ (𝐴 ↑_m 𝑛) ↔ ∃𝑛(𝑛 ∈ 𝐶 ∧ 𝐵 ∈ (𝐴 ↑_m 𝑛)))
10	8, 9	mpbir 231	1 ⊢ ∃*𝑛 ∈ 𝐶 𝐵 ∈ (𝐴 ↑_m 𝑛)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∃*wmo 2531 ∃*wrmo 3342 dom cdm 5619 ⟶wf 6478 (class class class)co 7349 ↑_m cmap 8753

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ral 3045 df-rex 3054 df-rmo 3343 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-fv 6490 df-ov 7352 df-oprab 7353 df-mpo 7354 df-1st 7924 df-2nd 7925 df-map 8755

This theorem is referenced by: (None)