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 6813 | . . . 4 ⊢ ((𝐹 Fn 𝐷 ∧ 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) → (𝐹 “ {𝐴, 𝐵}) = {(𝐹‘𝐴), (𝐹‘𝐵)}) | |
5 | 1, 2, 3, 4 | syl3anc 1373 | . . 3 ⊢ (𝜑 → (𝐹 “ {𝐴, 𝐵}) = {(𝐹‘𝐴), (𝐹‘𝐵)}) |
6 | fnimatp.4 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐷) | |
7 | fnsnfv 6808 | . . . . 5 ⊢ ((𝐹 Fn 𝐷 ∧ 𝐶 ∈ 𝐷) → {(𝐹‘𝐶)} = (𝐹 “ {𝐶})) | |
8 | 1, 6, 7 | syl2anc 587 | . . . 4 ⊢ (𝜑 → {(𝐹‘𝐶)} = (𝐹 “ {𝐶})) |
9 | 8 | eqcomd 2744 | . . 3 ⊢ (𝜑 → (𝐹 “ {𝐶}) = {(𝐹‘𝐶)}) |
10 | 5, 9 | uneq12d 4092 | . 2 ⊢ (𝜑 → ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶})) = ({(𝐹‘𝐴), (𝐹‘𝐵)} ∪ {(𝐹‘𝐶)})) |
11 | df-tp 4560 | . . . 4 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
12 | 11 | imaeq2i 5941 | . . 3 ⊢ (𝐹 “ {𝐴, 𝐵, 𝐶}) = (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶})) |
13 | imaundi 6027 | . . 3 ⊢ (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶})) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶})) | |
14 | 12, 13 | eqtri 2766 | . 2 ⊢ (𝐹 “ {𝐴, 𝐵, 𝐶}) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶})) |
15 | df-tp 4560 | . 2 ⊢ {(𝐹‘𝐴), (𝐹‘𝐵), (𝐹‘𝐶)} = ({(𝐹‘𝐴), (𝐹‘𝐵)} ∪ {(𝐹‘𝐶)}) | |
16 | 10, 14, 15 | 3eqtr4g 2804 | 1 ⊢ (𝜑 → (𝐹 “ {𝐴, 𝐵, 𝐶}) = {(𝐹‘𝐴), (𝐹‘𝐵), (𝐹‘𝐶)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2111 ∪ cun 3878 {csn 4555 {cpr 4557 {ctp 4559 “ cima 5568 Fn wfn 6392 ‘cfv 6397 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pr 5336 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3422 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-nul 4252 df-if 4454 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4834 df-br 5068 df-opab 5130 df-id 5469 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-iota 6355 df-fun 6399 df-fn 6400 df-fv 6405 |
This theorem is referenced by: s3rn 30964 |
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