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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 6156. (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 39894 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝑦}) | |
2 | iunab 4502 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝐹'''𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝑦} | |
3 | 1, 2 | syl6eqr 2662 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝐹'''𝑥) = 𝑦}) |
4 | df-sn 4126 | . . . . 5 ⊢ {(𝐹'''𝑥)} = {𝑦 ∣ 𝑦 = (𝐹'''𝑥)} | |
5 | eqcom 2617 | . . . . . 6 ⊢ (𝑦 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝑦) | |
6 | 5 | abbii 2726 | . . . . 5 ⊢ {𝑦 ∣ 𝑦 = (𝐹'''𝑥)} = {𝑦 ∣ (𝐹'''𝑥) = 𝑦} |
7 | 4, 6 | eqtri 2632 | . . . 4 ⊢ {(𝐹'''𝑥)} = {𝑦 ∣ (𝐹'''𝑥) = 𝑦} |
8 | 7 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐴 → {(𝐹'''𝑥)} = {𝑦 ∣ (𝐹'''𝑥) = 𝑦}) |
9 | 8 | iuneq2i 4475 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 {(𝐹'''𝑥)} = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝐹'''𝑥) = 𝑦} |
10 | 3, 9 | syl6eqr 2662 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {(𝐹'''𝑥)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {cab 2596 ∃wrex 2897 ⊆ wss 3540 {csn 4125 ∪ ciun 4455 dom cdm 5038 “ cima 5041 Fun wfun 5798 '''cafv 39843 |
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-iun 4457 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 df-dfat 39845 df-afv 39846 |
This theorem is referenced by: (None) |
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