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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 | funfvop 6237 | . . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹) | |
2 | ssel 3562 | . . . . . 6 ⊢ (𝐹 ⊆ 𝐺 → (〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹 → 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐺)) | |
3 | 1, 2 | syl5 33 | . . . . 5 ⊢ (𝐹 ⊆ 𝐺 → ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐺)) |
4 | 3 | imp 444 | . . . 4 ⊢ ((𝐹 ⊆ 𝐺 ∧ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)) → 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐺) |
5 | sneq 4135 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
6 | 5 | imaeq2d 5385 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐺 “ {𝑥}) = (𝐺 “ {𝐴})) |
7 | 6 | eleq2d 2673 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝐴) ∈ (𝐺 “ {𝑥}) ↔ (𝐹‘𝐴) ∈ (𝐺 “ {𝐴}))) |
8 | opeq1 4340 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → 〈𝑥, (𝐹‘𝐴)〉 = 〈𝐴, (𝐹‘𝐴)〉) | |
9 | 8 | eleq1d 2672 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (〈𝑥, (𝐹‘𝐴)〉 ∈ 𝐺 ↔ 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐺)) |
10 | vex 3176 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
11 | fvex 6113 | . . . . . . 7 ⊢ (𝐹‘𝐴) ∈ V | |
12 | 10, 11 | elimasn 5409 | . . . . . 6 ⊢ ((𝐹‘𝐴) ∈ (𝐺 “ {𝑥}) ↔ 〈𝑥, (𝐹‘𝐴)〉 ∈ 𝐺) |
13 | 7, 9, 12 | vtoclbg 3240 | . . . . 5 ⊢ (𝐴 ∈ dom 𝐹 → ((𝐹‘𝐴) ∈ (𝐺 “ {𝐴}) ↔ 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐺)) |
14 | 13 | ad2antll 761 | . . . 4 ⊢ ((𝐹 ⊆ 𝐺 ∧ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)) → ((𝐹‘𝐴) ∈ (𝐺 “ {𝐴}) ↔ 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐺)) |
15 | 4, 14 | mpbird 246 | . . 3 ⊢ ((𝐹 ⊆ 𝐺 ∧ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)) → (𝐹‘𝐴) ∈ (𝐺 “ {𝐴})) |
16 | 15 | exp32 629 | . 2 ⊢ (𝐹 ⊆ 𝐺 → (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) ∈ (𝐺 “ {𝐴})))) |
17 | 16 | impcom 445 | 1 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ 𝐺) → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) ∈ (𝐺 “ {𝐴}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 {csn 4125 〈cop 4131 dom cdm 5038 “ cima 5041 Fun wfun 5798 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 |
This theorem is referenced by: dfac3 8827 |
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