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| Mirrors > Home > MPE Home > Th. List > invdisjrab | Structured version Visualization version GIF version | ||
| Description: The restricted class abstractions {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦} for distinct 𝑦 ∈ 𝐴 are disjoint. (Contributed by AV, 6-May-2020.) (Proof shortened by GG, 26-Jan-2024.) |
| Ref | Expression |
|---|---|
| invdisjrab | ⊢ Disj 𝑦 ∈ 𝐴 {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑥𝑧 | |
| 2 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
| 3 | nfcsb1v 3923 | . . . . . 6 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐶 | |
| 4 | 3 | nfeq1 2921 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐶 = 𝑦 |
| 5 | csbeq1a 3913 | . . . . . 6 ⊢ (𝑥 = 𝑧 → 𝐶 = ⦋𝑧 / 𝑥⦌𝐶) | |
| 6 | 5 | eqeq1d 2739 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝐶 = 𝑦 ↔ ⦋𝑧 / 𝑥⦌𝐶 = 𝑦)) |
| 7 | 1, 2, 4, 6 | elrabf 3688 | . . . 4 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦} ↔ (𝑧 ∈ 𝐵 ∧ ⦋𝑧 / 𝑥⦌𝐶 = 𝑦)) |
| 8 | simprr 773 | . . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝐵 ∧ ⦋𝑧 / 𝑥⦌𝐶 = 𝑦)) → ⦋𝑧 / 𝑥⦌𝐶 = 𝑦) | |
| 9 | 7, 8 | sylan2b 594 | . . 3 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦}) → ⦋𝑧 / 𝑥⦌𝐶 = 𝑦) |
| 10 | 9 | rgen2 3199 | . 2 ⊢ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦}⦋𝑧 / 𝑥⦌𝐶 = 𝑦 |
| 11 | invdisj 5129 | . 2 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦}⦋𝑧 / 𝑥⦌𝐶 = 𝑦 → Disj 𝑦 ∈ 𝐴 {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦}) | |
| 12 | 10, 11 | ax-mp 5 | 1 ⊢ Disj 𝑦 ∈ 𝐴 {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦} |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 ⦋csb 3899 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-nfc 2892 df-ral 3062 df-rmo 3380 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-disj 5111 |
| This theorem is referenced by: disjwrdpfx 14738 |
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