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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 3448 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
2 | vex 3448 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
3 | 1, 2 | breldm 5865 | . . . . . 6 ⊢ (𝑦𝑅𝑥 → 𝑦 ∈ dom 𝑅) |
4 | 3 | pm4.71ri 562 | . . . . 5 ⊢ (𝑦𝑅𝑥 ↔ (𝑦 ∈ dom 𝑅 ∧ 𝑦𝑅𝑥)) |
5 | 4 | rexbii 3094 | . . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑦𝑅𝑥 ↔ ∃𝑦 ∈ 𝐴 (𝑦 ∈ dom 𝑅 ∧ 𝑦𝑅𝑥)) |
6 | rexin 4200 | . . . 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 6019 | . . 3 ⊢ (𝑥 ∈ (𝑅 “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑦𝑅𝑥) |
9 | 2 | elima 6019 | . . 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 2730 | 1 ⊢ (𝑅 “ 𝐴) = (𝑅 “ (𝐴 ∩ dom 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∃wrex 3070 ∩ cin 3910 class class class wbr 5106 dom cdm 5634 “ cima 5637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-xp 5640 df-cnv 5642 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 |
This theorem is referenced by: madeval2 27205 |
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