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Mirrors > Home > MPE Home > Th. List > Mathboxes > imaopab | Structured version Visualization version GIF version |
Description: The image of a class of ordered pairs. (Contributed by Steven Nguyen, 6-Jun-2023.) |
Ref | Expression |
---|---|
imaopab | ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ima 5682 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) = ran ({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) | |
2 | resopab 6024 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
3 | 2 | rneqi 5928 | . 2 ⊢ ran ({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) = ran {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
4 | rnopab 5945 | . . 3 ⊢ ran {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)} | |
5 | df-rex 3070 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
6 | 5 | abbii 2801 | . . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)} |
7 | 4, 6 | eqtr4i 2762 | . 2 ⊢ ran {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} |
8 | 1, 3, 7 | 3eqtri 2763 | 1 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 {cab 2708 ∃wrex 3069 {copab 5203 ran crn 5670 ↾ cres 5671 “ cima 5672 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-xp 5675 df-rel 5676 df-cnv 5677 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 |
This theorem is referenced by: prjspeclsp 41134 |
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