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| Mirrors > Home > MPE Home > Th. List > fnimatpd | 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 |
|---|---|
| fnimatpd.1 | ⊢ (𝜑 → 𝐹 Fn 𝐷) |
| fnimatpd.2 | ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
| fnimatpd.3 | ⊢ (𝜑 → 𝐵 ∈ 𝐷) |
| fnimatpd.4 | ⊢ (𝜑 → 𝐶 ∈ 𝐷) |
| Ref | Expression |
|---|---|
| fnimatpd | ⊢ (𝜑 → (𝐹 “ {𝐴, 𝐵, 𝐶}) = {(𝐹‘𝐴), (𝐹‘𝐵), (𝐹‘𝐶)}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fnimatpd.1 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐷) | |
| 2 | fnimatpd.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐷) | |
| 3 | fnimatpd.3 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
| 4 | fnimapr 6992 | . . . 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 | fnimatpd.4 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐷) | |
| 7 | fnsnfv 6988 | . . . . 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 2743 | . . 3 ⊢ (𝜑 → (𝐹 “ {𝐶}) = {(𝐹‘𝐶)}) |
| 10 | 5, 9 | uneq12d 4169 | . 2 ⊢ (𝜑 → ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶})) = ({(𝐹‘𝐴), (𝐹‘𝐵)} ∪ {(𝐹‘𝐶)})) |
| 11 | df-tp 4631 | . . . 4 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
| 12 | 11 | imaeq2i 6076 | . . 3 ⊢ (𝐹 “ {𝐴, 𝐵, 𝐶}) = (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶})) |
| 13 | imaundi 6169 | . . 3 ⊢ (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶})) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶})) | |
| 14 | 12, 13 | eqtri 2765 | . 2 ⊢ (𝐹 “ {𝐴, 𝐵, 𝐶}) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶})) |
| 15 | df-tp 4631 | . 2 ⊢ {(𝐹‘𝐴), (𝐹‘𝐵), (𝐹‘𝐶)} = ({(𝐹‘𝐴), (𝐹‘𝐵)} ∪ {(𝐹‘𝐶)}) | |
| 16 | 10, 14, 15 | 3eqtr4g 2802 | 1 ⊢ (𝜑 → (𝐹 “ {𝐴, 𝐵, 𝐶}) = {(𝐹‘𝐴), (𝐹‘𝐵), (𝐹‘𝐶)}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 {csn 4626 {cpr 4628 {ctp 4630 “ cima 5688 Fn wfn 6556 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 |
| This theorem is referenced by: s3rnOLD 32930 cycl3grtri 47914 grtrimap 47915 |
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