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Mirrors > Home > MPE Home > Th. List > invdisj | Structured version Visualization version GIF version |
Description: If there is a function 𝐶(𝑦) such that 𝐶(𝑦) = 𝑥 for all 𝑦 ∈ 𝐵(𝑥), then the sets 𝐵(𝑥) for distinct 𝑥 ∈ 𝐴 are disjoint. (Contributed by Mario Carneiro, 10-Dec-2016.) |
Ref | Expression |
---|---|
invdisj | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → Disj 𝑥 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfra2w 3287 | . . 3 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 | |
2 | df-ral 3052 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝐶 = 𝑥)) | |
3 | rsp 3235 | . . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → (𝑦 ∈ 𝐵 → 𝐶 = 𝑥)) | |
4 | eqcom 2733 | . . . . . . . . 9 ⊢ (𝐶 = 𝑥 ↔ 𝑥 = 𝐶) | |
5 | 3, 4 | imbitrdi 250 | . . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → (𝑦 ∈ 𝐵 → 𝑥 = 𝐶)) |
6 | 5 | imim2i 16 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝐶 = 𝑥) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑥 = 𝐶))) |
7 | 6 | impd 409 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝐶 = 𝑥) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝐶)) |
8 | 7 | alimi 1806 | . . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝐶 = 𝑥) → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝐶)) |
9 | 2, 8 | sylbi 216 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝐶)) |
10 | mo2icl 3708 | . . . 4 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝐶) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
12 | 1, 11 | alrimi 2202 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
13 | dfdisj2 5120 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
14 | 12, 13 | sylibr 233 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → Disj 𝑥 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∀wal 1532 = wceq 1534 ∈ wcel 2099 ∃*wmo 2527 ∀wral 3051 Disj wdisj 5118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rmo 3364 df-v 3464 df-disj 5119 |
This theorem is referenced by: invdisjrabw 5138 invdisjrab 5139 ackbijnn 15832 incexc2 15842 phisum 16792 itg1addlem1 25712 musum 27219 lgsquadlem1 27409 lgsquadlem2 27410 disjabrex 32502 disjabrexf 32503 actfunsnrndisj 34451 poimirlem27 37348 |
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