iunmapdisj - Metamath Proof Explorer

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Theorem iunmapdisj 10092

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 3729	. . . 4 ⊢ ∃*𝑛 𝑛 = dom 𝐵
2		elmapi 8907	. . . . . 6 ⊢ (𝐵 ∈ (𝐴 ↑_m 𝑛) → 𝐵:𝑛⟶𝐴)
3		fdm 6756	. . . . . . 7 ⊢ (𝐵:𝑛⟶𝐴 → dom 𝐵 = 𝑛)
4	3	eqcomd 2746	. . . . . 6 ⊢ (𝐵:𝑛⟶𝐴 → 𝑛 = dom 𝐵)
5	2, 4	syl 17	. . . . 5 ⊢ (𝐵 ∈ (𝐴 ↑_m 𝑛) → 𝑛 = dom 𝐵)
6	5	moimi 2548	. . . 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 2556	. 2 ⊢ ∃*𝑛(𝑛 ∈ 𝐶 ∧ 𝐵 ∈ (𝐴 ↑_m 𝑛))
9		df-rmo 3388	. 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 1537 ∈ wcel 2108 ∃*wmo 2541 ∃*wrmo 3387 dom cdm 5700 ⟶wf 6569 (class class class)co 7448 ↑_m cmap 8884

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ral 3068 df-rex 3077 df-rmo 3388 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-1st 8030 df-2nd 8031 df-map 8886

This theorem is referenced by: (None)