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Mirrors > Home > MPE Home > Th. List > iunmapdisj | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
iunmapdisj | ⊢ ∃*𝑛 ∈ 𝐶 𝐵 ∈ (𝐴 ↑𝑚 𝑛) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | moeq 3349 | . . . 4 ⊢ ∃*𝑛 𝑛 = dom 𝐵 | |
2 | elmapi 7765 | . . . . . 6 ⊢ (𝐵 ∈ (𝐴 ↑𝑚 𝑛) → 𝐵:𝑛⟶𝐴) | |
3 | fdm 5964 | . . . . . . 7 ⊢ (𝐵:𝑛⟶𝐴 → dom 𝐵 = 𝑛) | |
4 | 3 | eqcomd 2616 | . . . . . 6 ⊢ (𝐵:𝑛⟶𝐴 → 𝑛 = dom 𝐵) |
5 | 2, 4 | syl 17 | . . . . 5 ⊢ (𝐵 ∈ (𝐴 ↑𝑚 𝑛) → 𝑛 = dom 𝐵) |
6 | 5 | moimi 2508 | . . . 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 2513 | . 2 ⊢ ∃*𝑛(𝑛 ∈ 𝐶 ∧ 𝐵 ∈ (𝐴 ↑𝑚 𝑛)) |
9 | df-rmo 2904 | . 2 ⊢ (∃*𝑛 ∈ 𝐶 𝐵 ∈ (𝐴 ↑𝑚 𝑛) ↔ ∃*𝑛(𝑛 ∈ 𝐶 ∧ 𝐵 ∈ (𝐴 ↑𝑚 𝑛))) | |
10 | 8, 9 | mpbir 220 | 1 ⊢ ∃*𝑛 ∈ 𝐶 𝐵 ∈ (𝐴 ↑𝑚 𝑛) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃*wmo 2459 ∃*wrmo 2899 dom cdm 5038 ⟶wf 5800 (class class class)co 6549 ↑𝑚 cmap 7744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-map 7746 |
This theorem is referenced by: (None) |
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