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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 5969 | . . 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 1540 ∈ wcel 2108 ∃wrex 3070 ↦ cmpt 5225 ran crn 5686 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-mpt 5226 df-cnv 5693 df-dm 5695 df-rn 5696 | 
| This theorem is referenced by: cycsubggend 19223 nsgqusf1olem3 33443 zart0 33878 rr-elrnmpt3d 44221 | 
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