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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 3989 | . . . . 5 ⊢ (𝐹 ⊆ 𝐺 → (〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹 → 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐺)) | |
2 | funfvop 7070 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹) | |
3 | 1, 2 | impel 505 | . . . 4 ⊢ ((𝐹 ⊆ 𝐺 ∧ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)) → 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐺) |
4 | sneq 4641 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
5 | 4 | imaeq2d 6080 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐺 “ {𝑥}) = (𝐺 “ {𝐴})) |
6 | 5 | eleq2d 2825 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝐴) ∈ (𝐺 “ {𝑥}) ↔ (𝐹‘𝐴) ∈ (𝐺 “ {𝐴}))) |
7 | opeq1 4878 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → 〈𝑥, (𝐹‘𝐴)〉 = 〈𝐴, (𝐹‘𝐴)〉) | |
8 | 7 | eleq1d 2824 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (〈𝑥, (𝐹‘𝐴)〉 ∈ 𝐺 ↔ 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐺)) |
9 | vex 3482 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
10 | fvex 6920 | . . . . . . 7 ⊢ (𝐹‘𝐴) ∈ V | |
11 | 9, 10 | elimasn 6110 | . . . . . 6 ⊢ ((𝐹‘𝐴) ∈ (𝐺 “ {𝑥}) ↔ 〈𝑥, (𝐹‘𝐴)〉 ∈ 𝐺) |
12 | 6, 8, 11 | vtoclbg 3557 | . . . . 5 ⊢ (𝐴 ∈ dom 𝐹 → ((𝐹‘𝐴) ∈ (𝐺 “ {𝐴}) ↔ 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐺)) |
13 | 12 | ad2antll 729 | . . . 4 ⊢ ((𝐹 ⊆ 𝐺 ∧ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)) → ((𝐹‘𝐴) ∈ (𝐺 “ {𝐴}) ↔ 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐺)) |
14 | 3, 13 | mpbird 257 | . . 3 ⊢ ((𝐹 ⊆ 𝐺 ∧ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)) → (𝐹‘𝐴) ∈ (𝐺 “ {𝐴})) |
15 | 14 | exp32 420 | . 2 ⊢ (𝐹 ⊆ 𝐺 → (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) ∈ (𝐺 “ {𝐴})))) |
16 | 15 | impcom 407 | 1 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ 𝐺) → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) ∈ (𝐺 “ {𝐴}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 {csn 4631 〈cop 4637 dom cdm 5689 “ cima 5692 Fun wfun 6557 ‘cfv 6563 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-fv 6571 |
This theorem is referenced by: dfac3 10159 |
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