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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 3880 | . . . . 5 ⊢ (𝐹 ⊆ 𝐺 → (〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹 → 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐺)) | |
2 | funfvop 6839 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹) | |
3 | 1, 2 | impel 509 | . . . 4 ⊢ ((𝐹 ⊆ 𝐺 ∧ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)) → 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐺) |
4 | sneq 4536 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
5 | 4 | imaeq2d 5913 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐺 “ {𝑥}) = (𝐺 “ {𝐴})) |
6 | 5 | eleq2d 2819 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝐴) ∈ (𝐺 “ {𝑥}) ↔ (𝐹‘𝐴) ∈ (𝐺 “ {𝐴}))) |
7 | opeq1 4769 | . . . . . . 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 3404 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
10 | fvex 6699 | . . . . . . 7 ⊢ (𝐹‘𝐴) ∈ V | |
11 | 9, 10 | elimasn 5938 | . . . . . 6 ⊢ ((𝐹‘𝐴) ∈ (𝐺 “ {𝑥}) ↔ 〈𝑥, (𝐹‘𝐴)〉 ∈ 𝐺) |
12 | 6, 8, 11 | vtoclbg 3475 | . . . . 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 260 | . . 3 ⊢ ((𝐹 ⊆ 𝐺 ∧ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)) → (𝐹‘𝐴) ∈ (𝐺 “ {𝐴})) |
15 | 14 | exp32 424 | . 2 ⊢ (𝐹 ⊆ 𝐺 → (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) ∈ (𝐺 “ {𝐴})))) |
16 | 15 | impcom 411 | 1 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ 𝐺) → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) ∈ (𝐺 “ {𝐴}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ⊆ wss 3853 {csn 4526 〈cop 4532 dom cdm 5535 “ cima 5538 Fun wfun 6343 ‘cfv 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pr 5306 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3402 df-sbc 3686 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6307 df-fun 6351 df-fn 6352 df-fv 6357 |
This theorem is referenced by: dfac3 9633 |
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