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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rncnvepres | Structured version Visualization version GIF version | ||
| Description: The range of the restricted converse epsilon is the union of the restriction. (Contributed by Peter Mazsa, 11-Feb-2018.) (Revised by Peter Mazsa, 26-Sep-2021.) | 
| Ref | Expression | 
|---|---|
| rncnvepres | ⊢ ran (◡ E ↾ 𝐴) = ∪ 𝐴 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rnopab 5965 | . 2 ⊢ ran {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} | |
| 2 | cnvepres 38299 | . . 3 ⊢ (◡ E ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} | |
| 3 | 2 | rneqi 5948 | . 2 ⊢ ran (◡ E ↾ 𝐴) = ran {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} | 
| 4 | dfuni2 4909 | . . 3 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} | |
| 5 | df-rex 3071 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) | |
| 6 | 5 | abbii 2809 | . . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} | 
| 7 | 4, 6 | eqtri 2765 | . 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} | 
| 8 | 1, 3, 7 | 3eqtr4i 2775 | 1 ⊢ ran (◡ E ↾ 𝐴) = ∪ 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 ∃wrex 3070 ∪ cuni 4907 {copab 5205 E cep 5583 ◡ccnv 5684 ran crn 5686 ↾ cres 5687 | 
| 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-rab 3437 df-v 3482 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-br 5144 df-opab 5206 df-eprel 5584 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 | 
| This theorem is referenced by: dm1cosscnvepres 38457 | 
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