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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 3624 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐷 = 𝐵) |
4 | elrnmptdv.3 | . . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
5 | elrnmptdv.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
6 | 5 | elrnmpt 5821 | . . 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 259 | 1 ⊢ (𝜑 → 𝐷 ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∃wrex 3138 ↦ cmpt 5139 ran crn 5549 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-br 5060 df-opab 5122 df-mpt 5140 df-cnv 5556 df-dm 5558 df-rn 5559 |
This theorem is referenced by: cycsubggend 18341 rr-elrnmpt3d 40636 |
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