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Mathbox for Thierry Arnoux |
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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 31813 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}𝑦) | |
2 | eqid 2733 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 2 | rnmpt 5955 | . . 3 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} |
4 | disjeq1 5121 | . . 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 233 | 1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 {cab 2710 ∃wrex 3071 Disj wdisj 5114 ↦ cmpt 5232 ran crn 5678 |
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 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-cnv 5685 df-dm 5687 df-rn 5688 |
This theorem is referenced by: fnpreimac 31896 sigapildsys 33160 ldgenpisyslem1 33161 carsgclctunlem2 33318 pmeasadd 33324 |
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