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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fnimatp | Structured version Visualization version GIF version |
Description: The image of an unordered triple under a function. (Contributed by Thierry Arnoux, 19-Sep-2023.) |
Ref | Expression |
---|---|
fnimatp.1 | ⊢ (𝜑 → 𝐹 Fn 𝐷) |
fnimatp.2 | ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
fnimatp.3 | ⊢ (𝜑 → 𝐵 ∈ 𝐷) |
fnimatp.4 | ⊢ (𝜑 → 𝐶 ∈ 𝐷) |
Ref | Expression |
---|---|
fnimatp | ⊢ (𝜑 → (𝐹 “ {𝐴, 𝐵, 𝐶}) = {(𝐹‘𝐴), (𝐹‘𝐵), (𝐹‘𝐶)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnimatp.1 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐷) | |
2 | fnimatp.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐷) | |
3 | fnimatp.3 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
4 | fnimapr 6972 | . . . 4 ⊢ ((𝐹 Fn 𝐷 ∧ 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) → (𝐹 “ {𝐴, 𝐵}) = {(𝐹‘𝐴), (𝐹‘𝐵)}) | |
5 | 1, 2, 3, 4 | syl3anc 1371 | . . 3 ⊢ (𝜑 → (𝐹 “ {𝐴, 𝐵}) = {(𝐹‘𝐴), (𝐹‘𝐵)}) |
6 | fnimatp.4 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐷) | |
7 | fnsnfv 6967 | . . . . 5 ⊢ ((𝐹 Fn 𝐷 ∧ 𝐶 ∈ 𝐷) → {(𝐹‘𝐶)} = (𝐹 “ {𝐶})) | |
8 | 1, 6, 7 | syl2anc 584 | . . . 4 ⊢ (𝜑 → {(𝐹‘𝐶)} = (𝐹 “ {𝐶})) |
9 | 8 | eqcomd 2738 | . . 3 ⊢ (𝜑 → (𝐹 “ {𝐶}) = {(𝐹‘𝐶)}) |
10 | 5, 9 | uneq12d 4163 | . 2 ⊢ (𝜑 → ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶})) = ({(𝐹‘𝐴), (𝐹‘𝐵)} ∪ {(𝐹‘𝐶)})) |
11 | df-tp 4632 | . . . 4 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
12 | 11 | imaeq2i 6055 | . . 3 ⊢ (𝐹 “ {𝐴, 𝐵, 𝐶}) = (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶})) |
13 | imaundi 6146 | . . 3 ⊢ (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶})) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶})) | |
14 | 12, 13 | eqtri 2760 | . 2 ⊢ (𝐹 “ {𝐴, 𝐵, 𝐶}) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶})) |
15 | df-tp 4632 | . 2 ⊢ {(𝐹‘𝐴), (𝐹‘𝐵), (𝐹‘𝐶)} = ({(𝐹‘𝐴), (𝐹‘𝐵)} ∪ {(𝐹‘𝐶)}) | |
16 | 10, 14, 15 | 3eqtr4g 2797 | 1 ⊢ (𝜑 → (𝐹 “ {𝐴, 𝐵, 𝐶}) = {(𝐹‘𝐴), (𝐹‘𝐵), (𝐹‘𝐶)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ∪ cun 3945 {csn 4627 {cpr 4629 {ctp 4631 “ cima 5678 Fn wfn 6535 ‘cfv 6540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 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-tp 4632 df-op 4634 df-uni 4908 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 |
This theorem is referenced by: s3rn 32099 |
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