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Mirrors > Home > MPE Home > Th. List > elrnmptdv | Structured version Visualization version 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 3627 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐷 = 𝐵) |
4 | elrnmptdv.3 | . . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
5 | elrnmptdv.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
6 | 5 | elrnmpt 5972 | . . 3 ⊢ (𝐷 ∈ 𝑉 → (𝐷 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐷 = 𝐵)) |
7 | 4, 6 | syl 17 | . 2 ⊢ (𝜑 → (𝐷 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐷 = 𝐵)) |
8 | 3, 7 | mpbird 257 | 1 ⊢ (𝜑 → 𝐷 ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 ↦ cmpt 5231 ran crn 5690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-mpt 5232 df-cnv 5697 df-dm 5699 df-rn 5700 |
This theorem is referenced by: cycsubggend 19236 nsgqusf1olem3 33423 zart0 33840 rr-elrnmpt3d 44198 |
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