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Mirrors > Home > MPE Home > Th. List > Mathboxes > funimass4f | Structured version Visualization version GIF version |
Description: Membership relation for the values of a function whose image is a subclass. (Contributed by Thierry Arnoux, 24-Apr-2017.) |
Ref | Expression |
---|---|
funimass4f.1 | ⊢ Ⅎ𝑥𝐴 |
funimass4f.2 | ⊢ Ⅎ𝑥𝐵 |
funimass4f.3 | ⊢ Ⅎ𝑥𝐹 |
Ref | Expression |
---|---|
funimass4f | ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funimass4f.3 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 | |
2 | 1 | nffun 6525 | . . . . 5 ⊢ Ⅎ𝑥Fun 𝐹 |
3 | funimass4f.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
4 | 1 | nfdm 5907 | . . . . . 6 ⊢ Ⅎ𝑥dom 𝐹 |
5 | 3, 4 | nfss 3937 | . . . . 5 ⊢ Ⅎ𝑥 𝐴 ⊆ dom 𝐹 |
6 | 2, 5 | nfan 1903 | . . . 4 ⊢ Ⅎ𝑥(Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) |
7 | 1, 3 | nfima 6022 | . . . . 5 ⊢ Ⅎ𝑥(𝐹 “ 𝐴) |
8 | funimass4f.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
9 | 7, 8 | nfss 3937 | . . . 4 ⊢ Ⅎ𝑥(𝐹 “ 𝐴) ⊆ 𝐵 |
10 | 6, 9 | nfan 1903 | . . 3 ⊢ Ⅎ𝑥((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝐹 “ 𝐴) ⊆ 𝐵) |
11 | funfvima2 7182 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴))) | |
12 | ssel 3938 | . . . 4 ⊢ ((𝐹 “ 𝐴) ⊆ 𝐵 → ((𝐹‘𝑥) ∈ (𝐹 “ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)) | |
13 | 11, 12 | sylan9 509 | . . 3 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝐹 “ 𝐴) ⊆ 𝐵) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ 𝐵)) |
14 | 10, 13 | ralrimi 3239 | . 2 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝐹 “ 𝐴) ⊆ 𝐵) → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) |
15 | 3, 1 | dfimafnf 31596 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}) |
16 | 15 | adantr 482 | . . 3 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}) |
17 | 8 | abrexss 31481 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} ⊆ 𝐵) |
18 | 17 | adantl 483 | . . 3 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} ⊆ 𝐵) |
19 | 16, 18 | eqsstrd 3983 | . 2 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → (𝐹 “ 𝐴) ⊆ 𝐵) |
20 | 14, 19 | impbida 800 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {cab 2710 Ⅎwnfc 2884 ∀wral 3061 ∃wrex 3070 ⊆ wss 3911 dom cdm 5634 “ cima 5637 Fun wfun 6491 ‘cfv 6497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-fv 6505 |
This theorem is referenced by: ballotlem7 33192 |
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