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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 3477 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
2 | vex 3477 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
3 | 1, 2 | breldm 5915 | . . . . . 6 ⊢ (𝑦𝑅𝑥 → 𝑦 ∈ dom 𝑅) |
4 | 3 | pm4.71ri 559 | . . . . 5 ⊢ (𝑦𝑅𝑥 ↔ (𝑦 ∈ dom 𝑅 ∧ 𝑦𝑅𝑥)) |
5 | 4 | rexbii 3091 | . . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑦𝑅𝑥 ↔ ∃𝑦 ∈ 𝐴 (𝑦 ∈ dom 𝑅 ∧ 𝑦𝑅𝑥)) |
6 | rexin 4242 | . . . 4 ⊢ (∃𝑦 ∈ (𝐴 ∩ dom 𝑅)𝑦𝑅𝑥 ↔ ∃𝑦 ∈ 𝐴 (𝑦 ∈ dom 𝑅 ∧ 𝑦𝑅𝑥)) | |
7 | 5, 6 | bitr4i 277 | . . 3 ⊢ (∃𝑦 ∈ 𝐴 𝑦𝑅𝑥 ↔ ∃𝑦 ∈ (𝐴 ∩ dom 𝑅)𝑦𝑅𝑥) |
8 | 2 | elima 6073 | . . 3 ⊢ (𝑥 ∈ (𝑅 “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑦𝑅𝑥) |
9 | 2 | elima 6073 | . . 3 ⊢ (𝑥 ∈ (𝑅 “ (𝐴 ∩ dom 𝑅)) ↔ ∃𝑦 ∈ (𝐴 ∩ dom 𝑅)𝑦𝑅𝑥) |
10 | 7, 8, 9 | 3bitr4i 302 | . 2 ⊢ (𝑥 ∈ (𝑅 “ 𝐴) ↔ 𝑥 ∈ (𝑅 “ (𝐴 ∩ dom 𝑅))) |
11 | 10 | eqriv 2725 | 1 ⊢ (𝑅 “ 𝐴) = (𝑅 “ (𝐴 ∩ dom 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∃wrex 3067 ∩ cin 3948 class class class wbr 5152 dom cdm 5682 “ cima 5685 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-xp 5688 df-cnv 5690 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 |
This theorem is referenced by: madeval2 27800 |
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