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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 3582 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐷 = 𝐵) |
4 | elrnmptdv.3 | . . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
5 | elrnmptdv.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
6 | 5 | elrnmpt 5909 | . . 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 256 | 1 ⊢ (𝜑 → 𝐷 ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3071 ↦ cmpt 5186 ran crn 5632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-br 5104 df-opab 5166 df-mpt 5187 df-cnv 5639 df-dm 5641 df-rn 5642 |
This theorem is referenced by: cycsubggend 18989 nsgqusf1olem3 32076 zart0 32329 rr-elrnmpt3d 42423 |
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