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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 30332 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}𝑦) | |
2 | eqid 2821 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 2 | rnmpt 5827 | . . 3 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} |
4 | disjeq1 5038 | . . 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 236 | 1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 {cab 2799 ∃wrex 3139 Disj wdisj 5031 ↦ cmpt 5146 ran crn 5556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-disj 5032 df-br 5067 df-opab 5129 df-mpt 5147 df-cnv 5563 df-dm 5565 df-rn 5566 |
This theorem is referenced by: fnpreimac 30416 sigapildsys 31421 ldgenpisyslem1 31422 carsgclctunlem2 31577 pmeasadd 31583 |
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