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Mirrors > Home > MPE Home > Th. List > elrnmptd | Structured version Visualization version GIF version |
Description: The range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
elrnmptd.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
elrnmptd.x | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) |
elrnmptd.c | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
Ref | Expression |
---|---|
elrnmptd | ⊢ (𝜑 → 𝐶 ∈ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrnmptd.x | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) | |
2 | elrnmptd.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
3 | elrnmptd.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 3 | elrnmpt 5793 | . . 3 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)) |
5 | 2, 4 | syl 17 | . 2 ⊢ (𝜑 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)) |
6 | 1, 5 | mpbird 260 | 1 ⊢ (𝜑 → 𝐶 ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1542 ∈ wcel 2113 ∃wrex 3054 ↦ cmpt 5107 ran crn 5520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-rex 3059 df-v 3399 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-op 4520 df-br 5028 df-opab 5090 df-mpt 5108 df-cnv 5527 df-dm 5529 df-rn 5530 |
This theorem is referenced by: pwfilem 8768 nsgmgc 31161 nsgqusf1olem1 31162 zarclsun 31384 rnmptssrn 42241 infnsuprnmpt 42317 supminfrnmpt 42507 supminfxrrnmpt 42535 sge0sup 43455 sge0resplit 43470 sge0xaddlem2 43498 sge0pnfmpt 43509 sge0reuz 43511 sge0reuzb 43512 hoidmvlelem2 43660 |
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