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Mirrors > Home > MPE Home > Th. List > dffin1-5 | Structured version Visualization version GIF version |
Description: Compact quantifier-free version of the standard definition df-fin 8496. (Contributed by Stefan O'Rear, 6-Jan-2015.) |
Ref | Expression |
---|---|
dffin1-5 | ⊢ Fin = ( ≈ “ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ensymb 8540 | . . . 4 ⊢ (𝑥 ≈ 𝑦 ↔ 𝑦 ≈ 𝑥) | |
2 | 1 | rexbii 3210 | . . 3 ⊢ (∃𝑦 ∈ ω 𝑥 ≈ 𝑦 ↔ ∃𝑦 ∈ ω 𝑦 ≈ 𝑥) |
3 | 2 | abbii 2863 | . 2 ⊢ {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦} = {𝑥 ∣ ∃𝑦 ∈ ω 𝑦 ≈ 𝑥} |
4 | df-fin 8496 | . 2 ⊢ Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦} | |
5 | dfima2 5898 | . 2 ⊢ ( ≈ “ ω) = {𝑥 ∣ ∃𝑦 ∈ ω 𝑦 ≈ 𝑥} | |
6 | 3, 4, 5 | 3eqtr4i 2831 | 1 ⊢ Fin = ( ≈ “ ω) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 {cab 2776 ∃wrex 3107 class class class wbr 5030 “ cima 5522 ωcom 7560 ≈ cen 8489 Fincfn 8492 |
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-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 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-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-er 8272 df-en 8493 df-fin 8496 |
This theorem is referenced by: (None) |
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