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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.) |
Ref | Expression |
---|---|
invdisjrab | ⊢ Disj 𝑦 ∈ 𝐴 {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑥𝑧 | |
2 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑥𝐵 | |
3 | nfcsb1v 3919 | . . . . . . 7 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐶 | |
4 | 3 | nfeq1 2919 | . . . . . 6 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐶 = 𝑦 |
5 | csbeq1a 3908 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝐶 = ⦋𝑧 / 𝑥⦌𝐶) | |
6 | 5 | eqeq1d 2735 | . . . . . 6 ⊢ (𝑥 = 𝑧 → (𝐶 = 𝑦 ↔ ⦋𝑧 / 𝑥⦌𝐶 = 𝑦)) |
7 | 1, 2, 4, 6 | elrabf 3680 | . . . . 5 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦} ↔ (𝑧 ∈ 𝐵 ∧ ⦋𝑧 / 𝑥⦌𝐶 = 𝑦)) |
8 | ax-1 6 | . . . . 5 ⊢ (⦋𝑧 / 𝑥⦌𝐶 = 𝑦 → (𝑦 ∈ 𝐴 → ⦋𝑧 / 𝑥⦌𝐶 = 𝑦)) | |
9 | 7, 8 | simplbiim 506 | . . . 4 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦} → (𝑦 ∈ 𝐴 → ⦋𝑧 / 𝑥⦌𝐶 = 𝑦)) |
10 | 9 | impcom 409 | . . 3 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦}) → ⦋𝑧 / 𝑥⦌𝐶 = 𝑦) |
11 | 10 | rgen2 3198 | . 2 ⊢ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦}⦋𝑧 / 𝑥⦌𝐶 = 𝑦 |
12 | invdisj 5133 | . 2 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦}⦋𝑧 / 𝑥⦌𝐶 = 𝑦 → Disj 𝑦 ∈ 𝐴 {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦}) | |
13 | 11, 12 | ax-mp 5 | 1 ⊢ Disj 𝑦 ∈ 𝐴 {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∀wral 3062 {crab 3433 ⦋csb 3894 Disj wdisj 5114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rmo 3377 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-disj 5115 |
This theorem is referenced by: disjwrdpfx 14650 |
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