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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfaimafn2 | Structured version Visualization version GIF version |
Description: Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 6723. (Contributed by Alexander van der Vekens, 25-May-2017.) |
Ref | Expression |
---|---|
dfaimafn2 | ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {(𝐹'''𝑥)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfaimafn 43358 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝑦}) | |
2 | iunab 4967 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝐹'''𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝑦} | |
3 | 1, 2 | syl6eqr 2874 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝐹'''𝑥) = 𝑦}) |
4 | df-sn 4561 | . . . . 5 ⊢ {(𝐹'''𝑥)} = {𝑦 ∣ 𝑦 = (𝐹'''𝑥)} | |
5 | eqcom 2828 | . . . . . 6 ⊢ (𝑦 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝑦) | |
6 | 5 | abbii 2886 | . . . . 5 ⊢ {𝑦 ∣ 𝑦 = (𝐹'''𝑥)} = {𝑦 ∣ (𝐹'''𝑥) = 𝑦} |
7 | 4, 6 | eqtri 2844 | . . . 4 ⊢ {(𝐹'''𝑥)} = {𝑦 ∣ (𝐹'''𝑥) = 𝑦} |
8 | 7 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐴 → {(𝐹'''𝑥)} = {𝑦 ∣ (𝐹'''𝑥) = 𝑦}) |
9 | 8 | iuneq2i 4932 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 {(𝐹'''𝑥)} = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝐹'''𝑥) = 𝑦} |
10 | 3, 9 | syl6eqr 2874 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {(𝐹'''𝑥)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {cab 2799 ∃wrex 3139 ⊆ wss 3935 {csn 4560 ∪ ciun 4911 dom cdm 5549 “ cima 5552 Fun wfun 6343 '''cafv 43310 |
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 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-fal 1546 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-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-fv 6357 df-aiota 43279 df-dfat 43312 df-afv 43313 |
This theorem is referenced by: (None) |
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