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Mirrors > Home > MPE Home > Th. List > Mathboxes > elmptima | Structured version Visualization version GIF version |
Description: The image of a function in maps-to notation. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
elmptima | ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐷) ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐷)𝐶 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mptima 5934 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐷) = ran (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐵) | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝐶 ∈ 𝑉 → ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐷) = ran (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐵)) |
3 | 2 | eleq2d 2895 | . 2 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐷) ↔ 𝐶 ∈ ran (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐵))) |
4 | eqid 2818 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐵) = (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐵) | |
5 | 4 | elrnmpt 5821 | . 2 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ran (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐵) ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐷)𝐶 = 𝐵)) |
6 | 3, 5 | bitrd 280 | 1 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐷) ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐷)𝐶 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 ∃wrex 3136 ∩ cin 3932 ↦ cmpt 5137 ran crn 5549 “ cima 5551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-mpt 5138 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: liminfvalxr 41940 |
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