Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > dfimafnf | Structured version Visualization version GIF version |
Description: Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.) (Revised by Thierry Arnoux, 24-Apr-2017.) |
Ref | Expression |
---|---|
dfimafnf.1 | ⊢ Ⅎ𝑥𝐴 |
dfimafnf.2 | ⊢ Ⅎ𝑥𝐹 |
Ref | Expression |
---|---|
dfimafnf | ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3960 | . . . . . . 7 ⊢ (𝐴 ⊆ dom 𝐹 → (𝑧 ∈ 𝐴 → 𝑧 ∈ dom 𝐹)) | |
2 | eqcom 2828 | . . . . . . . . 9 ⊢ ((𝐹‘𝑧) = 𝑦 ↔ 𝑦 = (𝐹‘𝑧)) | |
3 | funbrfvb 6719 | . . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → ((𝐹‘𝑧) = 𝑦 ↔ 𝑧𝐹𝑦)) | |
4 | 2, 3 | syl5bbr 287 | . . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → (𝑦 = (𝐹‘𝑧) ↔ 𝑧𝐹𝑦)) |
5 | 4 | ex 415 | . . . . . . 7 ⊢ (Fun 𝐹 → (𝑧 ∈ dom 𝐹 → (𝑦 = (𝐹‘𝑧) ↔ 𝑧𝐹𝑦))) |
6 | 1, 5 | syl9r 78 | . . . . . 6 ⊢ (Fun 𝐹 → (𝐴 ⊆ dom 𝐹 → (𝑧 ∈ 𝐴 → (𝑦 = (𝐹‘𝑧) ↔ 𝑧𝐹𝑦)))) |
7 | 6 | imp31 420 | . . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ 𝑧 ∈ 𝐴) → (𝑦 = (𝐹‘𝑧) ↔ 𝑧𝐹𝑦)) |
8 | 7 | rexbidva 3296 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (∃𝑧 ∈ 𝐴 𝑦 = (𝐹‘𝑧) ↔ ∃𝑧 ∈ 𝐴 𝑧𝐹𝑦)) |
9 | 8 | abbidv 2885 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (𝐹‘𝑧)} = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑧𝐹𝑦}) |
10 | dfima2 5930 | . . 3 ⊢ (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑧𝐹𝑦} | |
11 | 9, 10 | syl6reqr 2875 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (𝐹‘𝑧)}) |
12 | nfcv 2977 | . . . 4 ⊢ Ⅎ𝑧𝐴 | |
13 | dfimafnf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
14 | dfimafnf.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 | |
15 | nfcv 2977 | . . . . . 6 ⊢ Ⅎ𝑥𝑧 | |
16 | 14, 15 | nffv 6679 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑧) |
17 | 16 | nfeq2 2995 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = (𝐹‘𝑧) |
18 | nfv 1911 | . . . 4 ⊢ Ⅎ𝑧 𝑦 = (𝐹‘𝑥) | |
19 | fveq2 6669 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) | |
20 | 19 | eqeq2d 2832 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝑦 = (𝐹‘𝑧) ↔ 𝑦 = (𝐹‘𝑥))) |
21 | 12, 13, 17, 18, 20 | cbvrexfw 3438 | . . 3 ⊢ (∃𝑧 ∈ 𝐴 𝑦 = (𝐹‘𝑧) ↔ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)) |
22 | 21 | abbii 2886 | . 2 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (𝐹‘𝑧)} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} |
23 | 11, 22 | syl6eq 2872 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {cab 2799 Ⅎwnfc 2961 ∃wrex 3139 ⊆ wss 3935 class class class wbr 5065 dom cdm 5554 “ cima 5557 Fun wfun 6348 ‘cfv 6354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-fv 6362 |
This theorem is referenced by: funimass4f 30381 |
Copyright terms: Public domain | W3C validator |