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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 1370 | . . 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 2741 | . . 3 ⊢ (𝜑 → (𝐹 “ {𝐶}) = {(𝐹‘𝐶)}) |
10 | 5, 9 | uneq12d 4179 | . 2 ⊢ (𝜑 → ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶})) = ({(𝐹‘𝐴), (𝐹‘𝐵)} ∪ {(𝐹‘𝐶)})) |
11 | df-tp 4636 | . . . 4 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
12 | 11 | imaeq2i 6078 | . . 3 ⊢ (𝐹 “ {𝐴, 𝐵, 𝐶}) = (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶})) |
13 | imaundi 6172 | . . 3 ⊢ (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶})) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶})) | |
14 | 12, 13 | eqtri 2763 | . 2 ⊢ (𝐹 “ {𝐴, 𝐵, 𝐶}) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶})) |
15 | df-tp 4636 | . 2 ⊢ {(𝐹‘𝐴), (𝐹‘𝐵), (𝐹‘𝐶)} = ({(𝐹‘𝐴), (𝐹‘𝐵)} ∪ {(𝐹‘𝐶)}) | |
16 | 10, 14, 15 | 3eqtr4g 2800 | 1 ⊢ (𝜑 → (𝐹 “ {𝐴, 𝐵, 𝐶}) = {(𝐹‘𝐴), (𝐹‘𝐵), (𝐹‘𝐶)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∪ cun 3961 {csn 4631 {cpr 4633 {ctp 4635 “ cima 5692 Fn wfn 6558 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-fv 6571 |
This theorem is referenced by: s3rnOLD 32915 grtrimap 47851 |
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