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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 6722 | . . . 4 ⊢ ((𝐹 Fn 𝐷 ∧ 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) → (𝐹 “ {𝐴, 𝐵}) = {(𝐹‘𝐴), (𝐹‘𝐵)}) | |
5 | 1, 2, 3, 4 | syl3anc 1368 | . . 3 ⊢ (𝜑 → (𝐹 “ {𝐴, 𝐵}) = {(𝐹‘𝐴), (𝐹‘𝐵)}) |
6 | fnimatp.4 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐷) | |
7 | fnsnfv 6718 | . . . . 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 2804 | . . 3 ⊢ (𝜑 → (𝐹 “ {𝐶}) = {(𝐹‘𝐶)}) |
10 | 5, 9 | uneq12d 4091 | . 2 ⊢ (𝜑 → ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶})) = ({(𝐹‘𝐴), (𝐹‘𝐵)} ∪ {(𝐹‘𝐶)})) |
11 | df-tp 4530 | . . . 4 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
12 | 11 | imaeq2i 5894 | . . 3 ⊢ (𝐹 “ {𝐴, 𝐵, 𝐶}) = (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶})) |
13 | imaundi 5975 | . . 3 ⊢ (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶})) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶})) | |
14 | 12, 13 | eqtri 2821 | . 2 ⊢ (𝐹 “ {𝐴, 𝐵, 𝐶}) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶})) |
15 | df-tp 4530 | . 2 ⊢ {(𝐹‘𝐴), (𝐹‘𝐵), (𝐹‘𝐶)} = ({(𝐹‘𝐴), (𝐹‘𝐵)} ∪ {(𝐹‘𝐶)}) | |
16 | 10, 14, 15 | 3eqtr4g 2858 | 1 ⊢ (𝜑 → (𝐹 “ {𝐴, 𝐵, 𝐶}) = {(𝐹‘𝐴), (𝐹‘𝐵), (𝐹‘𝐶)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∪ cun 3879 {csn 4525 {cpr 4527 {ctp 4529 “ cima 5522 Fn wfn 6319 ‘cfv 6324 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-fv 6332 |
This theorem is referenced by: s3rn 30648 |
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