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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 5563 | . . 3 ⊢ (𝐹 “ {𝑋}) = ran (𝐹 ↾ {𝑋}) | |
2 | 1 | eleq2i 2904 | . 2 ⊢ ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) ↔ (𝐹‘𝑋) ∈ ran (𝐹 ↾ {𝑋})) |
3 | opeq1 4797 | . . . . 5 ⊢ (𝑥 = 𝑋 → 〈𝑥, (𝐹‘𝑋)〉 = 〈𝑋, (𝐹‘𝑋)〉) | |
4 | 3 | eleq1d 2897 | . . . 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 6678 | . . . 4 ⊢ (𝐹‘𝑋) ∈ V | |
7 | elimasng 5950 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐹‘𝑋) ∈ V) → ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) ↔ 〈𝑋, (𝐹‘𝑋)〉 ∈ 𝐹)) | |
8 | 6, 7 | mpan2 689 | . . 3 ⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) ↔ 〈𝑋, (𝐹‘𝑋)〉 ∈ 𝐹)) |
9 | elrn2g 5756 | . . . 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 296 | . 2 ⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) → (𝐹‘𝑋) ∈ ran 𝐹)) |
12 | 2, 11 | syl5bir 245 | 1 ⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝐹‘𝑋) ∈ ran 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∃wex 1776 ∈ wcel 2110 Vcvv 3495 {csn 4561 〈cop 4567 ran crn 5551 ↾ cres 5552 “ cima 5553 ‘cfv 6350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-xp 5556 df-cnv 5558 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fv 6358 |
This theorem is referenced by: fvn0fvelrn 6920 funressndmfvrn 43272 |
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