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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 6355 | . . . . . 6 ⊢ (𝜑 → 𝐺 Fn dom 𝐺) |
6 | 5 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺 Fn dom 𝐺) |
7 | funimaeq.d | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐺) | |
8 | 7 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ dom 𝐺) |
9 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
10 | fnfvima 6973 | . . . . 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 2890 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐺 “ 𝐴)) |
13 | 1, 2, 12 | funimassd 41863 | . 2 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ (𝐺 “ 𝐴)) |
14 | 2 | funfnd 6355 | . . . . . 6 ⊢ (𝜑 → 𝐹 Fn dom 𝐹) |
15 | 14 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 Fn dom 𝐹) |
16 | funimaeq.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹) | |
17 | 16 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ dom 𝐹) |
18 | fnfvima 6973 | . . . . 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 2891 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐹 “ 𝐴)) |
21 | 1, 4, 20 | funimassd 41863 | . 2 ⊢ (𝜑 → (𝐺 “ 𝐴) ⊆ (𝐹 “ 𝐴)) |
22 | 13, 21 | eqssd 3932 | 1 ⊢ (𝜑 → (𝐹 “ 𝐴) = (𝐺 “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 ⊆ wss 3881 dom cdm 5519 “ cima 5522 Fun wfun 6318 Fn wfn 6319 ‘cfv 6324 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-fv 6332 |
This theorem is referenced by: (None) |
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