iunmapdisj - Metamath Proof Explorer

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Theorem iunmapdisj 10044

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 3695	. . . 4 ⊢ ∃*𝑛 𝑛 = dom 𝐵
2		elmapi 8864	. . . . . 6 ⊢ (𝐵 ∈ (𝐴 ↑_m 𝑛) → 𝐵:𝑛⟶𝐴)
3		fdm 6725	. . . . . . 7 ⊢ (𝐵:𝑛⟶𝐴 → dom 𝐵 = 𝑛)
4	3	eqcomd 2731	. . . . . 6 ⊢ (𝐵:𝑛⟶𝐴 → 𝑛 = dom 𝐵)
5	2, 4	syl 17	. . . . 5 ⊢ (𝐵 ∈ (𝐴 ↑_m 𝑛) → 𝑛 = dom 𝐵)
6	5	moimi 2533	. . . 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 2541	. 2 ⊢ ∃*𝑛(𝑛 ∈ 𝐶 ∧ 𝐵 ∈ (𝐴 ↑_m 𝑛))
9		df-rmo 3364	. 2 ⊢ (∃𝑛 ∈ 𝐶 𝐵 ∈ (𝐴 ↑_m 𝑛) ↔ ∃𝑛(𝑛 ∈ 𝐶 ∧ 𝐵 ∈ (𝐴 ↑_m 𝑛)))
10	8, 9	mpbir 230	1 ⊢ ∃*𝑛 ∈ 𝐶 𝐵 ∈ (𝐴 ↑_m 𝑛)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∃*wmo 2526 ∃*wrmo 3363 dom cdm 5672 ⟶wf 6538 (class class class)co 7415 ↑_m cmap 8841

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3052 df-rex 3061 df-rmo 3364 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-ov 7418 df-oprab 7419 df-mpo 7420 df-1st 7989 df-2nd 7990 df-map 8843

This theorem is referenced by: (None)