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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 3299 | . . 3 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 | |
| 2 | df-ral 3062 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝐶 = 𝑥)) | |
| 3 | rsp 3247 | . . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → (𝑦 ∈ 𝐵 → 𝐶 = 𝑥)) | |
| 4 | eqcom 2744 | . . . . . . . . 9 ⊢ (𝐶 = 𝑥 ↔ 𝑥 = 𝐶) | |
| 5 | 3, 4 | imbitrdi 251 | . . . . . . . 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 410 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝐶 = 𝑥) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝐶)) |
| 8 | 7 | alimi 1811 | . . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝐶 = 𝑥) → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝐶)) |
| 9 | 2, 8 | sylbi 217 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝐶)) |
| 10 | mo2icl 3720 | . . . 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 2213 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
| 13 | dfdisj2 5112 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
| 14 | 12, 13 | sylibr 234 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → Disj 𝑥 ∈ 𝐴 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ∃*wmo 2538 ∀wral 3061 Disj wdisj 5110 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rmo 3380 df-v 3482 df-disj 5111 |
| This theorem is referenced by: invdisjrab 5130 ackbijnn 15864 incexc2 15874 phisum 16828 itg1addlem1 25727 musum 27234 lgsquadlem1 27424 lgsquadlem2 27425 disjabrex 32595 disjabrexf 32596 actfunsnrndisj 34620 poimirlem27 37654 |
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