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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 5819 | . 2 ⊢ ran {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} | |
2 | cnvepres 35547 | . . 3 ⊢ (◡ E ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} | |
3 | 2 | rneqi 5800 | . 2 ⊢ ran (◡ E ↾ 𝐴) = ran {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} |
4 | dfuni2 4832 | . . 3 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} | |
5 | df-rex 3142 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) | |
6 | 5 | abbii 2884 | . . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} |
7 | 4, 6 | eqtri 2842 | . 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} |
8 | 1, 3, 7 | 3eqtr4i 2852 | 1 ⊢ ran (◡ E ↾ 𝐴) = ∪ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1531 ∃wex 1774 ∈ wcel 2108 {cab 2797 ∃wrex 3137 ∪ cuni 4830 {copab 5119 E cep 5457 ◡ccnv 5547 ran crn 5549 ↾ cres 5550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-eprel 5458 df-xp 5554 df-rel 5555 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 |
This theorem is referenced by: dm1cosscnvepres 35688 |
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