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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rr-elrnmpt3d | Structured version Visualization version GIF version |
Description: Elementhood in an image set. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
Ref | Expression |
---|---|
rr-elrnmpt3d.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
rr-elrnmpt3d.2 | ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
rr-elrnmpt3d.3 | ⊢ (𝜑 → 𝐷 ∈ 𝑉) |
rr-elrnmpt3d.4 | ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
rr-elrnmpt3d | ⊢ (𝜑 → 𝐷 ∈ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rr-elrnmpt3d.1 | . 2 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | rr-elrnmpt3d.2 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝐴) | |
3 | rr-elrnmpt3d.3 | . 2 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
4 | rr-elrnmpt3d.4 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐵 = 𝐷) | |
5 | 4 | eqcomd 2731 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐷 = 𝐵) |
6 | 1, 2, 3, 5 | elrnmptdv 5959 | 1 ⊢ (𝜑 → 𝐷 ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ↦ cmpt 5226 ran crn 5673 |
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-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
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-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5144 df-opab 5206 df-mpt 5227 df-cnv 5680 df-dm 5682 df-rn 5683 |
This theorem is referenced by: mnurndlem1 43783 |
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