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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 3961 | . . . . 5 ⊢ (𝐹 ⊆ 𝐺 → (〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹 → 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐺)) | |
2 | funfvop 6820 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹) | |
3 | 1, 2 | impel 508 | . . . 4 ⊢ ((𝐹 ⊆ 𝐺 ∧ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)) → 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐺) |
4 | sneq 4577 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
5 | 4 | imaeq2d 5929 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐺 “ {𝑥}) = (𝐺 “ {𝐴})) |
6 | 5 | eleq2d 2898 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝐴) ∈ (𝐺 “ {𝑥}) ↔ (𝐹‘𝐴) ∈ (𝐺 “ {𝐴}))) |
7 | opeq1 4803 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → 〈𝑥, (𝐹‘𝐴)〉 = 〈𝐴, (𝐹‘𝐴)〉) | |
8 | 7 | eleq1d 2897 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (〈𝑥, (𝐹‘𝐴)〉 ∈ 𝐺 ↔ 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐺)) |
9 | vex 3497 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
10 | fvex 6683 | . . . . . . 7 ⊢ (𝐹‘𝐴) ∈ V | |
11 | 9, 10 | elimasn 5954 | . . . . . 6 ⊢ ((𝐹‘𝐴) ∈ (𝐺 “ {𝑥}) ↔ 〈𝑥, (𝐹‘𝐴)〉 ∈ 𝐺) |
12 | 6, 8, 11 | vtoclbg 3569 | . . . . 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 259 | . . 3 ⊢ ((𝐹 ⊆ 𝐺 ∧ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)) → (𝐹‘𝐴) ∈ (𝐺 “ {𝐴})) |
15 | 14 | exp32 423 | . 2 ⊢ (𝐹 ⊆ 𝐺 → (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) ∈ (𝐺 “ {𝐴})))) |
16 | 15 | impcom 410 | 1 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ 𝐺) → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) ∈ (𝐺 “ {𝐴}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 {csn 4567 〈cop 4573 dom cdm 5555 “ cima 5558 Fun wfun 6349 ‘cfv 6355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-fv 6363 |
This theorem is referenced by: dfac3 9547 |
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