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Mirrors > Home > MPE Home > Th. List > funfvima3 | Structured version Visualization version GIF version |
Description: A class including a function contains the function's value in the image of the singleton of the argument. (Contributed by NM, 23-Mar-2004.) |
Ref | Expression |
---|---|
funfvima3 | ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ 𝐺) → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) ∈ (𝐺 “ {𝐴}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3975 | . . . . 5 ⊢ (𝐹 ⊆ 𝐺 → (⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹 → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐺)) | |
2 | funfvop 7051 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹) | |
3 | 1, 2 | impel 506 | . . . 4 ⊢ ((𝐹 ⊆ 𝐺 ∧ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)) → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐺) |
4 | sneq 4638 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
5 | 4 | imaeq2d 6059 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐺 “ {𝑥}) = (𝐺 “ {𝐴})) |
6 | 5 | eleq2d 2819 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝐴) ∈ (𝐺 “ {𝑥}) ↔ (𝐹‘𝐴) ∈ (𝐺 “ {𝐴}))) |
7 | opeq1 4873 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → ⟨𝑥, (𝐹‘𝐴)⟩ = ⟨𝐴, (𝐹‘𝐴)⟩) | |
8 | 7 | eleq1d 2818 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (⟨𝑥, (𝐹‘𝐴)⟩ ∈ 𝐺 ↔ ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐺)) |
9 | vex 3478 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
10 | fvex 6904 | . . . . . . 7 ⊢ (𝐹‘𝐴) ∈ V | |
11 | 9, 10 | elimasn 6088 | . . . . . 6 ⊢ ((𝐹‘𝐴) ∈ (𝐺 “ {𝑥}) ↔ ⟨𝑥, (𝐹‘𝐴)⟩ ∈ 𝐺) |
12 | 6, 8, 11 | vtoclbg 3559 | . . . . 5 ⊢ (𝐴 ∈ dom 𝐹 → ((𝐹‘𝐴) ∈ (𝐺 “ {𝐴}) ↔ ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐺)) |
13 | 12 | ad2antll 727 | . . . 4 ⊢ ((𝐹 ⊆ 𝐺 ∧ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)) → ((𝐹‘𝐴) ∈ (𝐺 “ {𝐴}) ↔ ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐺)) |
14 | 3, 13 | mpbird 256 | . . 3 ⊢ ((𝐹 ⊆ 𝐺 ∧ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)) → (𝐹‘𝐴) ∈ (𝐺 “ {𝐴})) |
15 | 14 | exp32 421 | . 2 ⊢ (𝐹 ⊆ 𝐺 → (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) ∈ (𝐺 “ {𝐴})))) |
16 | 15 | impcom 408 | 1 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ 𝐺) → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) ∈ (𝐺 “ {𝐴}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⊆ wss 3948 {csn 4628 ⟨cop 4634 dom cdm 5676 “ cima 5679 Fun wfun 6537 ‘cfv 6543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-fv 6551 |
This theorem is referenced by: dfac3 10118 |
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