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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 6952. (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 45808 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝑦}) | |
2 | iunab 5053 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝐹'''𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝑦} | |
3 | 1, 2 | eqtr4di 2791 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝐹'''𝑥) = 𝑦}) |
4 | df-sn 4628 | . . . . 5 ⊢ {(𝐹'''𝑥)} = {𝑦 ∣ 𝑦 = (𝐹'''𝑥)} | |
5 | eqcom 2740 | . . . . . 6 ⊢ (𝑦 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝑦) | |
6 | 5 | abbii 2803 | . . . . 5 ⊢ {𝑦 ∣ 𝑦 = (𝐹'''𝑥)} = {𝑦 ∣ (𝐹'''𝑥) = 𝑦} |
7 | 4, 6 | eqtri 2761 | . . . 4 ⊢ {(𝐹'''𝑥)} = {𝑦 ∣ (𝐹'''𝑥) = 𝑦} |
8 | 7 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐴 → {(𝐹'''𝑥)} = {𝑦 ∣ (𝐹'''𝑥) = 𝑦}) |
9 | 8 | iuneq2i 5017 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 {(𝐹'''𝑥)} = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝐹'''𝑥) = 𝑦} |
10 | 3, 9 | eqtr4di 2791 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {(𝐹'''𝑥)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {cab 2710 ∃wrex 3071 ⊆ wss 3947 {csn 4627 ∪ ciun 4996 dom cdm 5675 “ cima 5678 Fun wfun 6534 '''cafv 45760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-fv 6548 df-aiota 45728 df-dfat 45762 df-afv 45763 |
This theorem is referenced by: (None) |
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