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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 3635 | . . . 4 ⊢ ∃*𝑛 𝑛 = dom 𝐵 | |
2 | elmapi 8281 | . . . . . 6 ⊢ (𝐵 ∈ (𝐴 ↑𝑚 𝑛) → 𝐵:𝑛⟶𝐴) | |
3 | fdm 6393 | . . . . . . 7 ⊢ (𝐵:𝑛⟶𝐴 → dom 𝐵 = 𝑛) | |
4 | 3 | eqcomd 2800 | . . . . . 6 ⊢ (𝐵:𝑛⟶𝐴 → 𝑛 = dom 𝐵) |
5 | 2, 4 | syl 17 | . . . . 5 ⊢ (𝐵 ∈ (𝐴 ↑𝑚 𝑛) → 𝑛 = dom 𝐵) |
6 | 5 | moimi 2580 | . . . 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 2592 | . 2 ⊢ ∃*𝑛(𝑛 ∈ 𝐶 ∧ 𝐵 ∈ (𝐴 ↑𝑚 𝑛)) |
9 | df-rmo 3112 | . 2 ⊢ (∃*𝑛 ∈ 𝐶 𝐵 ∈ (𝐴 ↑𝑚 𝑛) ↔ ∃*𝑛(𝑛 ∈ 𝐶 ∧ 𝐵 ∈ (𝐴 ↑𝑚 𝑛))) | |
10 | 8, 9 | mpbir 232 | 1 ⊢ ∃*𝑛 ∈ 𝐶 𝐵 ∈ (𝐴 ↑𝑚 𝑛) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1522 ∈ wcel 2080 ∃*wmo 2573 ∃*wrmo 3107 dom cdm 5446 ⟶wf 6224 (class class class)co 7019 ↑𝑚 cmap 8259 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-ral 3109 df-rex 3110 df-rmo 3112 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-op 4481 df-uni 4748 df-iun 4829 df-br 4965 df-opab 5027 df-mpt 5044 df-id 5351 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-fv 6236 df-ov 7022 df-oprab 7023 df-mpo 7024 df-1st 7548 df-2nd 7549 df-map 8261 |
This theorem is referenced by: (None) |
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