iunmapdisj - Metamath Proof Explorer

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Theorem iunmapdisj 10007

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 3679	. . . 4 ⊢ ∃*𝑛 𝑛 = dom 𝐵
2		elmapi 8846	. . . . . 6 ⊢ (𝐵 ∈ (𝐴 ↑_m 𝑛) → 𝐵:𝑛⟶𝐴)
3		fdm 6716	. . . . . . 7 ⊢ (𝐵:𝑛⟶𝐴 → dom 𝐵 = 𝑛)
4	3	eqcomd 2775	. . . . . 6 ⊢ (𝐵:𝑛⟶𝐴 → 𝑛 = dom 𝐵)
5	2, 4	syl 18	. . . . 5 ⊢ (𝐵 ∈ (𝐴 ↑_m 𝑛) → 𝑛 = dom 𝐵)
6	5	moimi 2579	. . . 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 2587	. 2 ⊢ ∃*𝑛(𝑛 ∈ 𝐶 ∧ 𝐵 ∈ (𝐴 ↑_m 𝑛))
9		df-rmo 3376	. 2 ⊢ (∃𝑛 ∈ 𝐶 𝐵 ∈ (𝐴 ↑_m 𝑛) ↔ ∃𝑛(𝑛 ∈ 𝐶 ∧ 𝐵 ∈ (𝐴 ↑_m 𝑛)))
10	8, 9	mpbir 234	1 ⊢ ∃*𝑛 ∈ 𝐶 𝐵 ∈ (𝐴 ↑_m 𝑛)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 400 = wceq 1567 ∈ wcel 2149 ∃*wmo 2571 ∃*wrmo 3375 dom cdm 5662 ⟶wf 6533 (class class class)co 7411 ↑_m cmap 8824

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-ral 3086 df-rex 3096 df-rmo 3376 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-id 5557 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7986 df-2nd 7987 df-map 8826

This theorem is referenced by: (None)