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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 8525 | . . 3 ⊢ 1o ≠ ∅ | |
3 | 2 | necomi 2993 | . 2 ⊢ ∅ ≠ 1o |
4 | fvpr1g 7210 | . 2 ⊢ ((∅ ∈ ω ∧ 𝐴 ∈ 𝑉 ∧ ∅ ≠ 1o) → ({〈∅, 𝐴〉, 〈1o, 𝐵〉}‘∅) = 𝐴) | |
5 | 1, 3, 4 | mp3an13 1451 | 1 ⊢ (𝐴 ∈ 𝑉 → ({〈∅, 𝐴〉, 〈1o, 𝐵〉}‘∅) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∅c0 4339 {cpr 4633 〈cop 4637 ‘cfv 6563 ωcom 7887 1oc1o 8498 |
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-11 2155 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-3or 1087 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-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-res 5701 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fv 6571 df-om 7888 df-1o 8505 |
This theorem is referenced by: fvprif 17608 xpsfeq 17610 xpsfrnel2 17611 xpsff1o 17614 xpsle 17626 dmdprdpr 20084 dprdpr 20085 xpstopnlem1 23833 xpstopnlem2 23835 xpsxmetlem 24405 xpsdsval 24407 xpsmet 24408 |
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