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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 6569 | . . . . . 6 ⊢ (𝜑 → 𝐺 Fn dom 𝐺) |
6 | 5 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺 Fn dom 𝐺) |
7 | funimaeq.d | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐺) | |
8 | 7 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ dom 𝐺) |
9 | simpr 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
10 | fnfvima 7220 | . . . . 5 ⊢ ((𝐺 Fn dom 𝐺 ∧ 𝐴 ⊆ dom 𝐺 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐺 “ 𝐴)) | |
11 | 6, 8, 9, 10 | syl3anc 1371 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐺 “ 𝐴)) |
12 | 3, 11 | eqeltrd 2833 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐺 “ 𝐴)) |
13 | 1, 2, 12 | funimassd 43765 | . 2 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ (𝐺 “ 𝐴)) |
14 | 2 | funfnd 6569 | . . . . . 6 ⊢ (𝜑 → 𝐹 Fn dom 𝐹) |
15 | 14 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 Fn dom 𝐹) |
16 | funimaeq.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹) | |
17 | 16 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ dom 𝐹) |
18 | fnfvima 7220 | . . . . 5 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐴 ⊆ dom 𝐹 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴)) | |
19 | 15, 17, 9, 18 | syl3anc 1371 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴)) |
20 | 3, 19 | eqeltrrd 2834 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐹 “ 𝐴)) |
21 | 1, 4, 20 | funimassd 43765 | . 2 ⊢ (𝜑 → (𝐺 “ 𝐴) ⊆ (𝐹 “ 𝐴)) |
22 | 13, 21 | eqssd 3996 | 1 ⊢ (𝜑 → (𝐹 “ 𝐴) = (𝐺 “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 Ⅎwnf 1785 ∈ wcel 2106 ⊆ wss 3945 dom cdm 5670 “ cima 5673 Fun wfun 6527 Fn wfn 6528 ‘cfv 6533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pr 5421 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5143 df-opab 5205 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-iota 6485 df-fun 6535 df-fn 6536 df-fv 6541 |
This theorem is referenced by: (None) |
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