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Mirrors > Home > MPE Home > Th. List > imaindm | Structured version Visualization version GIF version |
Description: The image is unaffected by intersection with the domain. (Contributed by Scott Fenton, 17-Dec-2021.) |
Ref | Expression |
---|---|
imaindm | ⊢ (𝑅 “ 𝐴) = (𝑅 “ (𝐴 ∩ dom 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3473 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
2 | vex 3473 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
3 | 1, 2 | breldm 5905 | . . . . . 6 ⊢ (𝑦𝑅𝑥 → 𝑦 ∈ dom 𝑅) |
4 | 3 | pm4.71ri 560 | . . . . 5 ⊢ (𝑦𝑅𝑥 ↔ (𝑦 ∈ dom 𝑅 ∧ 𝑦𝑅𝑥)) |
5 | 4 | rexbii 3089 | . . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑦𝑅𝑥 ↔ ∃𝑦 ∈ 𝐴 (𝑦 ∈ dom 𝑅 ∧ 𝑦𝑅𝑥)) |
6 | rexin 4235 | . . . 4 ⊢ (∃𝑦 ∈ (𝐴 ∩ dom 𝑅)𝑦𝑅𝑥 ↔ ∃𝑦 ∈ 𝐴 (𝑦 ∈ dom 𝑅 ∧ 𝑦𝑅𝑥)) | |
7 | 5, 6 | bitr4i 278 | . . 3 ⊢ (∃𝑦 ∈ 𝐴 𝑦𝑅𝑥 ↔ ∃𝑦 ∈ (𝐴 ∩ dom 𝑅)𝑦𝑅𝑥) |
8 | 2 | elima 6062 | . . 3 ⊢ (𝑥 ∈ (𝑅 “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑦𝑅𝑥) |
9 | 2 | elima 6062 | . . 3 ⊢ (𝑥 ∈ (𝑅 “ (𝐴 ∩ dom 𝑅)) ↔ ∃𝑦 ∈ (𝐴 ∩ dom 𝑅)𝑦𝑅𝑥) |
10 | 7, 8, 9 | 3bitr4i 303 | . 2 ⊢ (𝑥 ∈ (𝑅 “ 𝐴) ↔ 𝑥 ∈ (𝑅 “ (𝐴 ∩ dom 𝑅))) |
11 | 10 | eqriv 2724 | 1 ⊢ (𝑅 “ 𝐴) = (𝑅 “ (𝐴 ∩ dom 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∃wrex 3065 ∩ cin 3943 class class class wbr 5142 dom cdm 5672 “ cima 5675 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-xp 5678 df-cnv 5680 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 |
This theorem is referenced by: madeval2 27754 |
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