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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 5561 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) = ran ({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) | |
2 | resopab 5895 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
3 | 2 | rneqi 5800 | . 2 ⊢ ran ({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) = ran {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
4 | rnopab 5819 | . . 3 ⊢ ran {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)} | |
5 | df-rex 3143 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
6 | 5 | abbii 2885 | . . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)} |
7 | 4, 6 | eqtr4i 2846 | . 2 ⊢ ran {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} |
8 | 1, 3, 7 | 3eqtri 2847 | 1 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1536 ∃wex 1779 ∈ wcel 2113 {cab 2798 ∃wrex 3138 {copab 5121 ran crn 5549 ↾ cres 5550 “ cima 5551 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-br 5060 df-opab 5122 df-xp 5554 df-rel 5555 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 |
This theorem is referenced by: prjspeclsp 39339 |
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