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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 5578 | . . 3 ⊢ (𝐹 “ {𝑋}) = ran (𝐹 ↾ {𝑋}) | |
2 | 1 | eleq2i 2830 | . 2 ⊢ ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) ↔ (𝐹‘𝑋) ∈ ran (𝐹 ↾ {𝑋})) |
3 | opeq1 4798 | . . . . 5 ⊢ (𝑥 = 𝑋 → 〈𝑥, (𝐹‘𝑋)〉 = 〈𝑋, (𝐹‘𝑋)〉) | |
4 | 3 | eleq1d 2823 | . . . 4 ⊢ (𝑥 = 𝑋 → (〈𝑥, (𝐹‘𝑋)〉 ∈ 𝐹 ↔ 〈𝑋, (𝐹‘𝑋)〉 ∈ 𝐹)) |
5 | 4 | spcegv 3524 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (〈𝑋, (𝐹‘𝑋)〉 ∈ 𝐹 → ∃𝑥〈𝑥, (𝐹‘𝑋)〉 ∈ 𝐹)) |
6 | fvex 6748 | . . . 4 ⊢ (𝐹‘𝑋) ∈ V | |
7 | elimasng 5970 | . . . 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 5773 | . . . 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 297 | . 2 ⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) → (𝐹‘𝑋) ∈ ran 𝐹)) |
12 | 2, 11 | syl5bir 246 | 1 ⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝐹‘𝑋) ∈ ran 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∃wex 1787 ∈ wcel 2111 Vcvv 3420 {csn 4555 〈cop 4561 ran crn 5566 ↾ cres 5567 “ cima 5568 ‘cfv 6397 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pr 5336 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3422 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-nul 4252 df-if 4454 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4834 df-br 5068 df-opab 5130 df-xp 5571 df-cnv 5573 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-iota 6355 df-fv 6405 |
This theorem is referenced by: fvn0fvelrn 6996 funressndmfvrn 44238 |
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