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 3562 | . . . . . . 7 ⊢ (𝐴 ⊆ dom 𝐹 → (𝑧 ∈ 𝐴 → 𝑧 ∈ dom 𝐹)) | |
2 | eqcom 2617 | . . . . . . . . 9 ⊢ ((𝐹‘𝑧) = 𝑦 ↔ 𝑦 = (𝐹‘𝑧)) | |
3 | funbrfvb 6148 | . . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → ((𝐹‘𝑧) = 𝑦 ↔ 𝑧𝐹𝑦)) | |
4 | 2, 3 | syl5bbr 273 | . . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → (𝑦 = (𝐹‘𝑧) ↔ 𝑧𝐹𝑦)) |
5 | 4 | ex 449 | . . . . . . 7 ⊢ (Fun 𝐹 → (𝑧 ∈ dom 𝐹 → (𝑦 = (𝐹‘𝑧) ↔ 𝑧𝐹𝑦))) |
6 | 1, 5 | syl9r 76 | . . . . . 6 ⊢ (Fun 𝐹 → (𝐴 ⊆ dom 𝐹 → (𝑧 ∈ 𝐴 → (𝑦 = (𝐹‘𝑧) ↔ 𝑧𝐹𝑦)))) |
7 | 6 | imp31 447 | . . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ 𝑧 ∈ 𝐴) → (𝑦 = (𝐹‘𝑧) ↔ 𝑧𝐹𝑦)) |
8 | 7 | rexbidva 3031 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (∃𝑧 ∈ 𝐴 𝑦 = (𝐹‘𝑧) ↔ ∃𝑧 ∈ 𝐴 𝑧𝐹𝑦)) |
9 | 8 | abbidv 2728 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (𝐹‘𝑧)} = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑧𝐹𝑦}) |
10 | dfima2 5387 | . . 3 ⊢ (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑧𝐹𝑦} | |
11 | 9, 10 | syl6reqr 2663 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (𝐹‘𝑧)}) |
12 | nfcv 2751 | . . . 4 ⊢ Ⅎ𝑧𝐴 | |
13 | dfimafnf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
14 | dfimafnf.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 | |
15 | nfcv 2751 | . . . . . 6 ⊢ Ⅎ𝑥𝑧 | |
16 | 14, 15 | nffv 6110 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑧) |
17 | 16 | nfeq2 2766 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = (𝐹‘𝑧) |
18 | nfv 1830 | . . . 4 ⊢ Ⅎ𝑧 𝑦 = (𝐹‘𝑥) | |
19 | fveq2 6103 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) | |
20 | 19 | eqeq2d 2620 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝑦 = (𝐹‘𝑧) ↔ 𝑦 = (𝐹‘𝑥))) |
21 | 12, 13, 17, 18, 20 | cbvrexf 3142 | . . 3 ⊢ (∃𝑧 ∈ 𝐴 𝑦 = (𝐹‘𝑧) ↔ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)) |
22 | 21 | abbii 2726 | . 2 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (𝐹‘𝑧)} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} |
23 | 11, 22 | syl6eq 2660 | 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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {cab 2596 Ⅎwnfc 2738 ∃wrex 2897 ⊆ wss 3540 class class class wbr 4583 dom cdm 5038 “ cima 5041 Fun wfun 5798 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 |
This theorem is referenced by: funimass4f 28818 |
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