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Mathbox for Thierry Arnoux |
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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 3746 | . . . . . . 7 ⊢ (𝐴 ⊆ dom 𝐹 → (𝑧 ∈ 𝐴 → 𝑧 ∈ dom 𝐹)) | |
2 | eqcom 2778 | . . . . . . . . 9 ⊢ ((𝐹‘𝑧) = 𝑦 ↔ 𝑦 = (𝐹‘𝑧)) | |
3 | funbrfvb 6379 | . . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → ((𝐹‘𝑧) = 𝑦 ↔ 𝑧𝐹𝑦)) | |
4 | 2, 3 | syl5bbr 274 | . . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → (𝑦 = (𝐹‘𝑧) ↔ 𝑧𝐹𝑦)) |
5 | 4 | ex 397 | . . . . . . 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 404 | . . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ 𝑧 ∈ 𝐴) → (𝑦 = (𝐹‘𝑧) ↔ 𝑧𝐹𝑦)) |
8 | 7 | rexbidva 3197 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (∃𝑧 ∈ 𝐴 𝑦 = (𝐹‘𝑧) ↔ ∃𝑧 ∈ 𝐴 𝑧𝐹𝑦)) |
9 | 8 | abbidv 2890 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (𝐹‘𝑧)} = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑧𝐹𝑦}) |
10 | dfima2 5609 | . . 3 ⊢ (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑧𝐹𝑦} | |
11 | 9, 10 | syl6reqr 2824 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (𝐹‘𝑧)}) |
12 | nfcv 2913 | . . . 4 ⊢ Ⅎ𝑧𝐴 | |
13 | dfimafnf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
14 | dfimafnf.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 | |
15 | nfcv 2913 | . . . . . 6 ⊢ Ⅎ𝑥𝑧 | |
16 | 14, 15 | nffv 6339 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑧) |
17 | 16 | nfeq2 2929 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = (𝐹‘𝑧) |
18 | nfv 1995 | . . . 4 ⊢ Ⅎ𝑧 𝑦 = (𝐹‘𝑥) | |
19 | fveq2 6332 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) | |
20 | 19 | eqeq2d 2781 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝑦 = (𝐹‘𝑧) ↔ 𝑦 = (𝐹‘𝑥))) |
21 | 12, 13, 17, 18, 20 | cbvrexf 3315 | . . 3 ⊢ (∃𝑧 ∈ 𝐴 𝑦 = (𝐹‘𝑧) ↔ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)) |
22 | 21 | abbii 2888 | . 2 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (𝐹‘𝑧)} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} |
23 | 11, 22 | syl6eq 2821 | 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 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 {cab 2757 Ⅎwnfc 2900 ∃wrex 3062 ⊆ wss 3723 class class class wbr 4786 dom cdm 5249 “ cima 5252 Fun wfun 6025 ‘cfv 6031 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-fv 6039 |
This theorem is referenced by: funimass4f 29777 |
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