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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 3472 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
2 | vex 3472 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
3 | 1, 2 | breldm 5901 | . . . . . 6 ⊢ (𝑦𝑅𝑥 → 𝑦 ∈ dom 𝑅) |
4 | 3 | pm4.71ri 560 | . . . . 5 ⊢ (𝑦𝑅𝑥 ↔ (𝑦 ∈ dom 𝑅 ∧ 𝑦𝑅𝑥)) |
5 | 4 | rexbii 3088 | . . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑦𝑅𝑥 ↔ ∃𝑦 ∈ 𝐴 (𝑦 ∈ dom 𝑅 ∧ 𝑦𝑅𝑥)) |
6 | rexin 4234 | . . . 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 6057 | . . 3 ⊢ (𝑥 ∈ (𝑅 “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑦𝑅𝑥) |
9 | 2 | elima 6057 | . . 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 2723 | 1 ⊢ (𝑅 “ 𝐴) = (𝑅 “ (𝐴 ∩ dom 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∃wrex 3064 ∩ cin 3942 class class class wbr 5141 dom cdm 5669 “ cima 5672 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-xp 5675 df-cnv 5677 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 |
This theorem is referenced by: madeval2 27730 |
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