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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > funimaeq | Structured version Visualization version GIF version |
Description: Membership relation for the values of a function whose image is a subclass. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
funimaeq.x | ⊢ Ⅎ𝑥𝜑 |
funimaeq.f | ⊢ (𝜑 → Fun 𝐹) |
funimaeq.g | ⊢ (𝜑 → Fun 𝐺) |
funimaeq.a | ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹) |
funimaeq.d | ⊢ (𝜑 → 𝐴 ⊆ dom 𝐺) |
funimaeq.e | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥)) |
Ref | Expression |
---|---|
funimaeq | ⊢ (𝜑 → (𝐹 “ 𝐴) = (𝐺 “ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funimaeq.x | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | funimaeq.f | . . 3 ⊢ (𝜑 → Fun 𝐹) | |
3 | funimaeq.e | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥)) | |
4 | funimaeq.g | . . . . . . 7 ⊢ (𝜑 → Fun 𝐺) | |
5 | 4 | funfnd 6589 | . . . . . 6 ⊢ (𝜑 → 𝐺 Fn dom 𝐺) |
6 | 5 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺 Fn dom 𝐺) |
7 | funimaeq.d | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐺) | |
8 | 7 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ dom 𝐺) |
9 | simpr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
10 | fnfvima 7251 | . . . . 5 ⊢ ((𝐺 Fn dom 𝐺 ∧ 𝐴 ⊆ dom 𝐺 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐺 “ 𝐴)) | |
11 | 6, 8, 9, 10 | syl3anc 1368 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐺 “ 𝐴)) |
12 | 3, 11 | eqeltrd 2829 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐺 “ 𝐴)) |
13 | 1, 2, 12 | funimassd 6970 | . 2 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ (𝐺 “ 𝐴)) |
14 | 2 | funfnd 6589 | . . . . . 6 ⊢ (𝜑 → 𝐹 Fn dom 𝐹) |
15 | 14 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 Fn dom 𝐹) |
16 | funimaeq.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹) | |
17 | 16 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ dom 𝐹) |
18 | fnfvima 7251 | . . . . 5 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐴 ⊆ dom 𝐹 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴)) | |
19 | 15, 17, 9, 18 | syl3anc 1368 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴)) |
20 | 3, 19 | eqeltrrd 2830 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐹 “ 𝐴)) |
21 | 1, 4, 20 | funimassd 6970 | . 2 ⊢ (𝜑 → (𝐺 “ 𝐴) ⊆ (𝐹 “ 𝐴)) |
22 | 13, 21 | eqssd 3999 | 1 ⊢ (𝜑 → (𝐹 “ 𝐴) = (𝐺 “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 ⊆ wss 3949 dom cdm 5682 “ cima 5685 Fun wfun 6547 Fn wfn 6548 ‘cfv 6553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-fv 6561 |
This theorem is referenced by: (None) |
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