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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 6379 | . . . . . 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 6986 | . . . . 5 ⊢ ((𝐺 Fn dom 𝐺 ∧ 𝐴 ⊆ dom 𝐺 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐺 “ 𝐴)) | |
11 | 6, 8, 9, 10 | syl3anc 1363 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐺 “ 𝐴)) |
12 | 3, 11 | eqeltrd 2910 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐺 “ 𝐴)) |
13 | 1, 2, 12 | funimassd 41373 | . 2 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ (𝐺 “ 𝐴)) |
14 | 2 | funfnd 6379 | . . . . . 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 6986 | . . . . 5 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐴 ⊆ dom 𝐹 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴)) | |
19 | 15, 17, 9, 18 | syl3anc 1363 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴)) |
20 | 3, 19 | eqeltrrd 2911 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐹 “ 𝐴)) |
21 | 1, 4, 20 | funimassd 41373 | . 2 ⊢ (𝜑 → (𝐺 “ 𝐴) ⊆ (𝐹 “ 𝐴)) |
22 | 13, 21 | eqssd 3981 | 1 ⊢ (𝜑 → (𝐹 “ 𝐴) = (𝐺 “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 Ⅎwnf 1775 ∈ wcel 2105 ⊆ wss 3933 dom cdm 5548 “ cima 5551 Fun wfun 6342 Fn wfn 6343 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-fv 6356 |
This theorem is referenced by: (None) |
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