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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 5561 | . . 3 ⊢ (𝐹 “ {𝑋}) = ran (𝐹 ↾ {𝑋}) | |
2 | 1 | eleq2i 2901 | . 2 ⊢ ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) ↔ (𝐹‘𝑋) ∈ ran (𝐹 ↾ {𝑋})) |
3 | opeq1 4795 | . . . . 5 ⊢ (𝑥 = 𝑋 → 〈𝑥, (𝐹‘𝑋)〉 = 〈𝑋, (𝐹‘𝑋)〉) | |
4 | 3 | eleq1d 2894 | . . . 4 ⊢ (𝑥 = 𝑋 → (〈𝑥, (𝐹‘𝑋)〉 ∈ 𝐹 ↔ 〈𝑋, (𝐹‘𝑋)〉 ∈ 𝐹)) |
5 | 4 | spcegv 3594 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (〈𝑋, (𝐹‘𝑋)〉 ∈ 𝐹 → ∃𝑥〈𝑥, (𝐹‘𝑋)〉 ∈ 𝐹)) |
6 | fvex 6676 | . . . 4 ⊢ (𝐹‘𝑋) ∈ V | |
7 | elimasng 5948 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐹‘𝑋) ∈ V) → ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) ↔ 〈𝑋, (𝐹‘𝑋)〉 ∈ 𝐹)) | |
8 | 6, 7 | mpan2 687 | . . 3 ⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) ↔ 〈𝑋, (𝐹‘𝑋)〉 ∈ 𝐹)) |
9 | elrn2g 5754 | . . . 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 295 | . 2 ⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) → (𝐹‘𝑋) ∈ ran 𝐹)) |
12 | 2, 11 | syl5bir 244 | 1 ⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝐹‘𝑋) ∈ ran 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∃wex 1771 ∈ wcel 2105 Vcvv 3492 {csn 4557 〈cop 4563 ran crn 5549 ↾ cres 5550 “ cima 5551 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fv 6356 |
This theorem is referenced by: fvn0fvelrn 6917 funressndmfvrn 43156 |
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