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| Mirrors > Home > MPE Home > Th. List > fvpr0o | Structured version Visualization version GIF version | ||
| Description: The value of a function with a domain of (at most) two elements. (Contributed by Jim Kingdon, 25-Sep-2023.) | 
| Ref | Expression | 
|---|---|
| fvpr0o | ⊢ (𝐴 ∈ 𝑉 → ({〈∅, 𝐴〉, 〈1o, 𝐵〉}‘∅) = 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | peano1 7911 | . 2 ⊢ ∅ ∈ ω | |
| 2 | 1n0 8527 | . . 3 ⊢ 1o ≠ ∅ | |
| 3 | 2 | necomi 2994 | . 2 ⊢ ∅ ≠ 1o | 
| 4 | fvpr1g 7211 | . 2 ⊢ ((∅ ∈ ω ∧ 𝐴 ∈ 𝑉 ∧ ∅ ≠ 1o) → ({〈∅, 𝐴〉, 〈1o, 𝐵〉}‘∅) = 𝐴) | |
| 5 | 1, 3, 4 | mp3an13 1453 | 1 ⊢ (𝐴 ∈ 𝑉 → ({〈∅, 𝐴〉, 〈1o, 𝐵〉}‘∅) = 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∅c0 4332 {cpr 4627 〈cop 4631 ‘cfv 6560 ωcom 7888 1oc1o 8500 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-res 5696 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fv 6568 df-om 7889 df-1o 8507 | 
| This theorem is referenced by: fvprif 17607 xpsfeq 17609 xpsfrnel2 17610 xpsff1o 17613 xpsle 17625 dmdprdpr 20070 dprdpr 20071 xpstopnlem1 23818 xpstopnlem2 23820 xpsxmetlem 24390 xpsdsval 24392 xpsmet 24393 | 
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