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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 2739 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐷 = 𝐵) |
6 | 1, 2, 3, 5 | elrnmptdv 5921 | 1 ⊢ (𝜑 → 𝐷 ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ↦ cmpt 5192 ran crn 5638 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
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-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-mpt 5193 df-cnv 5645 df-dm 5647 df-rn 5648 |
This theorem is referenced by: mnurndlem1 42653 |
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