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Mirrors > Home > ILE Home > Th. List > elrnmptdv | GIF version |
Description: Elementhood in the range of a function in maps-to notation, deduction form. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
Ref | Expression |
---|---|
elrnmptdv.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
elrnmptdv.2 | ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
elrnmptdv.3 | ⊢ (𝜑 → 𝐷 ∈ 𝑉) |
elrnmptdv.4 | ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐷 = 𝐵) |
Ref | Expression |
---|---|
elrnmptdv | ⊢ (𝜑 → 𝐷 ∈ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrnmptdv.4 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐷 = 𝐵) | |
2 | elrnmptdv.2 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐴) | |
3 | 1, 2 | rspcime 2832 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐷 = 𝐵) |
4 | elrnmptdv.3 | . . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
5 | elrnmptdv.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
6 | 5 | elrnmpt 4847 | . . 3 ⊢ (𝐷 ∈ 𝑉 → (𝐷 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐷 = 𝐵)) |
7 | 4, 6 | syl 14 | . 2 ⊢ (𝜑 → (𝐷 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐷 = 𝐵)) |
8 | 3, 7 | mpbird 166 | 1 ⊢ (𝜑 → 𝐷 ∈ ran 𝐹) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1342 ∈ wcel 2135 ∃wrex 2443 ↦ cmpt 4037 ran crn 4599 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-mpt 4039 df-cnv 4606 df-dm 4608 df-rn 4609 |
This theorem is referenced by: (None) |
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