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Mirrors > Home > MPE Home > Th. List > Mathboxes > dffv5 | Structured version Visualization version GIF version |
Description: Another quantifier-free definition of function value. (Contributed by Scott Fenton, 19-Feb-2013.) |
Ref | Expression |
---|---|
dffv5 | ⊢ (𝐹‘𝐴) = ∪ ∪ ({(𝐹 “ {𝐴})} ∩ Singletons ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffv3 6641 | . 2 ⊢ (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) | |
2 | dfiota3 33497 | . 2 ⊢ (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = ∪ ∪ ({{𝑥 ∣ 𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons ) | |
3 | abid2 2932 | . . . . . 6 ⊢ {𝑥 ∣ 𝑥 ∈ (𝐹 “ {𝐴})} = (𝐹 “ {𝐴}) | |
4 | 3 | sneqi 4536 | . . . . 5 ⊢ {{𝑥 ∣ 𝑥 ∈ (𝐹 “ {𝐴})}} = {(𝐹 “ {𝐴})} |
5 | 4 | ineq1i 4135 | . . . 4 ⊢ ({{𝑥 ∣ 𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons ) = ({(𝐹 “ {𝐴})} ∩ Singletons ) |
6 | 5 | unieqi 4813 | . . 3 ⊢ ∪ ({{𝑥 ∣ 𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons ) = ∪ ({(𝐹 “ {𝐴})} ∩ Singletons ) |
7 | 6 | unieqi 4813 | . 2 ⊢ ∪ ∪ ({{𝑥 ∣ 𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons ) = ∪ ∪ ({(𝐹 “ {𝐴})} ∩ Singletons ) |
8 | 1, 2, 7 | 3eqtri 2825 | 1 ⊢ (𝐹‘𝐴) = ∪ ∪ ({(𝐹 “ {𝐴})} ∩ Singletons ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 {cab 2776 ∩ cin 3880 {csn 4525 ∪ cuni 4800 “ cima 5522 ℩cio 6281 ‘cfv 6324 Singletons csingles 33413 |
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-pow 5231 ax-pr 5295 ax-un 7441 |
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-symdif 4169 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-eprel 5430 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-f 6328 df-fo 6330 df-fv 6332 df-1st 7671 df-2nd 7672 df-txp 33428 df-singleton 33436 df-singles 33437 |
This theorem is referenced by: brapply 33512 |
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