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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 5568 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) = ran ({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) | |
2 | resopab 5902 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
3 | 2 | rneqi 5807 | . 2 ⊢ ran ({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) = ran {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
4 | rnopab 5826 | . . 3 ⊢ ran {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)} | |
5 | df-rex 3144 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
6 | 5 | abbii 2886 | . . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)} |
7 | 4, 6 | eqtr4i 2847 | . 2 ⊢ ran {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} |
8 | 1, 3, 7 | 3eqtri 2848 | 1 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 {cab 2799 ∃wrex 3139 {copab 5128 ran crn 5556 ↾ cres 5557 “ cima 5558 |
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-rex 3144 df-rab 3147 df-v 3496 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-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 |
This theorem is referenced by: prjspeclsp 39311 |
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