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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 6659 | . 2 ⊢ (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) | |
2 | dfiota3 33281 | . 2 ⊢ (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = ∪ ∪ ({{𝑥 ∣ 𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons ) | |
3 | abid2 2954 | . . . . . 6 ⊢ {𝑥 ∣ 𝑥 ∈ (𝐹 “ {𝐴})} = (𝐹 “ {𝐴}) | |
4 | 3 | sneqi 4568 | . . . . 5 ⊢ {{𝑥 ∣ 𝑥 ∈ (𝐹 “ {𝐴})}} = {(𝐹 “ {𝐴})} |
5 | 4 | ineq1i 4182 | . . . 4 ⊢ ({{𝑥 ∣ 𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons ) = ({(𝐹 “ {𝐴})} ∩ Singletons ) |
6 | 5 | unieqi 4839 | . . 3 ⊢ ∪ ({{𝑥 ∣ 𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons ) = ∪ ({(𝐹 “ {𝐴})} ∩ Singletons ) |
7 | 6 | unieqi 4839 | . 2 ⊢ ∪ ∪ ({{𝑥 ∣ 𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons ) = ∪ ∪ ({(𝐹 “ {𝐴})} ∩ Singletons ) |
8 | 1, 2, 7 | 3eqtri 2845 | 1 ⊢ (𝐹‘𝐴) = ∪ ∪ ({(𝐹 “ {𝐴})} ∩ Singletons ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 {cab 2796 ∩ cin 3932 {csn 4557 ∪ cuni 4830 “ cima 5551 ℩cio 6305 ‘cfv 6348 Singletons csingles 33197 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-symdif 4216 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-eprel 5458 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fo 6354 df-fv 6356 df-1st 7678 df-2nd 7679 df-txp 33212 df-singleton 33220 df-singles 33221 |
This theorem is referenced by: brapply 33296 |
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