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| Mirrors > Home > MPE Home > Th. List > Mathboxes > disjrnmpt | Structured version Visualization version GIF version | ||
| Description: Rewriting a disjoint collection using the range of a mapping. (Contributed by Thierry Arnoux, 27-May-2020.) |
| Ref | Expression |
|---|---|
| disjrnmpt | ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | disjabrex 32595 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}𝑦) | |
| 2 | eqid 2737 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 3 | 2 | rnmpt 5968 | . . 3 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} |
| 4 | disjeq1 5117 | . . 3 ⊢ (ran (𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} → (Disj 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ↔ Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}𝑦)) | |
| 5 | 3, 4 | ax-mp 5 | . 2 ⊢ (Disj 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ↔ Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}𝑦) |
| 6 | 1, 5 | sylibr 234 | 1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 {cab 2714 ∃wrex 3070 Disj wdisj 5110 ↦ cmpt 5225 ran crn 5686 |
| 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 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-disj 5111 df-br 5144 df-opab 5206 df-mpt 5226 df-cnv 5693 df-dm 5695 df-rn 5696 |
| This theorem is referenced by: fnpreimac 32681 sigapildsys 34163 ldgenpisyslem1 34164 carsgclctunlem2 34321 pmeasadd 34327 |
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