Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 5814 | . . 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 259 | 1 ⊢ (𝜑 → 𝐶 ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 ↦ cmpt 5132 ran crn 5542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pr 5316 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-rab 3147 df-v 3488 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-br 5053 df-opab 5115 df-mpt 5133 df-cnv 5549 df-dm 5551 df-rn 5552 |
This theorem is referenced by: rnmptssrn 41532 infnsuprnmpt 41612 supminfrnmpt 41809 supminfxrrnmpt 41837 sge0sup 42763 sge0resplit 42778 sge0xaddlem2 42806 sge0pnfmpt 42817 sge0reuz 42819 sge0reuzb 42820 hoidmvlelem2 42968 |
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