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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 3613 | . . . 4 ⊢ ∃*𝑛 𝑛 = dom 𝐵 | |
2 | elmapi 8519 | . . . . . 6 ⊢ (𝐵 ∈ (𝐴 ↑m 𝑛) → 𝐵:𝑛⟶𝐴) | |
3 | fdm 6543 | . . . . . . 7 ⊢ (𝐵:𝑛⟶𝐴 → dom 𝐵 = 𝑛) | |
4 | 3 | eqcomd 2740 | . . . . . 6 ⊢ (𝐵:𝑛⟶𝐴 → 𝑛 = dom 𝐵) |
5 | 2, 4 | syl 17 | . . . . 5 ⊢ (𝐵 ∈ (𝐴 ↑m 𝑛) → 𝑛 = dom 𝐵) |
6 | 5 | moimi 2542 | . . . 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 2550 | . 2 ⊢ ∃*𝑛(𝑛 ∈ 𝐶 ∧ 𝐵 ∈ (𝐴 ↑m 𝑛)) |
9 | df-rmo 3062 | . 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 399 = wceq 1543 ∈ wcel 2110 ∃*wmo 2535 ∃*wrmo 3057 dom cdm 5540 ⟶wf 6365 (class class class)co 7202 ↑m cmap 8497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-fv 6377 df-ov 7205 df-oprab 7206 df-mpo 7207 df-1st 7750 df-2nd 7751 df-map 8499 |
This theorem is referenced by: (None) |
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