iunmapdisj - Metamath Proof Explorer

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Theorem iunmapdisj 9983

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 3681	. . . 4 ⊢ ∃*𝑛 𝑛 = dom 𝐵
2		elmapi 8825	. . . . . 6 ⊢ (𝐵 ∈ (𝐴 ↑_m 𝑛) → 𝐵:𝑛⟶𝐴)
3		fdm 6700	. . . . . . 7 ⊢ (𝐵:𝑛⟶𝐴 → dom 𝐵 = 𝑛)
4	3	eqcomd 2736	. . . . . 6 ⊢ (𝐵:𝑛⟶𝐴 → 𝑛 = dom 𝐵)
5	2, 4	syl 17	. . . . 5 ⊢ (𝐵 ∈ (𝐴 ↑_m 𝑛) → 𝑛 = dom 𝐵)
6	5	moimi 2539	. . . 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 2547	. 2 ⊢ ∃*𝑛(𝑛 ∈ 𝐶 ∧ 𝐵 ∈ (𝐴 ↑_m 𝑛))
9		df-rmo 3356	. 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 2532 ∃*wrmo 3355 dom cdm 5641 ⟶wf 6510 (class class class)co 7390 ↑_m cmap 8802

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 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ral 3046 df-rex 3055 df-rmo 3356 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-fv 6522 df-ov 7393 df-oprab 7394 df-mpo 7395 df-1st 7971 df-2nd 7972 df-map 8804

This theorem is referenced by: (None)