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| Mirrors > Home > MPE Home > Th. List > fvrnressn | Structured version Visualization version GIF version | ||
| Description: If the value of a function is in the range of the function restricted to the singleton containing the argument, then the value of the function is in the range of the function. (Contributed by Alexander van der Vekens, 22-Jul-2018.) |
| Ref | Expression |
|---|---|
| fvrnressn | ⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝐹‘𝑋) ∈ ran 𝐹)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-ima 5698 | . . 3 ⊢ (𝐹 “ {𝑋}) = ran (𝐹 ↾ {𝑋}) | |
| 2 | 1 | eleq2i 2833 | . 2 ⊢ ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) ↔ (𝐹‘𝑋) ∈ ran (𝐹 ↾ {𝑋})) |
| 3 | opeq1 4873 | . . . . 5 ⊢ (𝑥 = 𝑋 → 〈𝑥, (𝐹‘𝑋)〉 = 〈𝑋, (𝐹‘𝑋)〉) | |
| 4 | 3 | eleq1d 2826 | . . . 4 ⊢ (𝑥 = 𝑋 → (〈𝑥, (𝐹‘𝑋)〉 ∈ 𝐹 ↔ 〈𝑋, (𝐹‘𝑋)〉 ∈ 𝐹)) |
| 5 | 4 | spcegv 3597 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (〈𝑋, (𝐹‘𝑋)〉 ∈ 𝐹 → ∃𝑥〈𝑥, (𝐹‘𝑋)〉 ∈ 𝐹)) |
| 6 | fvex 6919 | . . . 4 ⊢ (𝐹‘𝑋) ∈ V | |
| 7 | elimasng 6107 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐹‘𝑋) ∈ V) → ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) ↔ 〈𝑋, (𝐹‘𝑋)〉 ∈ 𝐹)) | |
| 8 | 6, 7 | mpan2 691 | . . 3 ⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) ↔ 〈𝑋, (𝐹‘𝑋)〉 ∈ 𝐹)) |
| 9 | elrn2g 5901 | . . . 4 ⊢ ((𝐹‘𝑋) ∈ V → ((𝐹‘𝑋) ∈ ran 𝐹 ↔ ∃𝑥〈𝑥, (𝐹‘𝑋)〉 ∈ 𝐹)) | |
| 10 | 6, 9 | mp1i 13 | . . 3 ⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ ran 𝐹 ↔ ∃𝑥〈𝑥, (𝐹‘𝑋)〉 ∈ 𝐹)) |
| 11 | 5, 8, 10 | 3imtr4d 294 | . 2 ⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) → (𝐹‘𝑋) ∈ ran 𝐹)) |
| 12 | 2, 11 | biimtrrid 243 | 1 ⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝐹‘𝑋) ∈ ran 𝐹)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 {csn 4626 〈cop 4632 ran crn 5686 ↾ cres 5687 “ cima 5688 ‘cfv 6561 |
| 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-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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fv 6569 |
| This theorem is referenced by: fvn0fvelrnOLD 7183 funressndmfvrn 47056 |
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