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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 7601 | . 2 ⊢ ∅ ∈ ω | |
2 | 1n0 8119 | . . 3 ⊢ 1o ≠ ∅ | |
3 | 2 | necomi 3070 | . 2 ⊢ ∅ ≠ 1o |
4 | fvpr1g 6954 | . 2 ⊢ ((∅ ∈ ω ∧ 𝐴 ∈ 𝑉 ∧ ∅ ≠ 1o) → ({〈∅, 𝐴〉, 〈1o, 𝐵〉}‘∅) = 𝐴) | |
5 | 1, 3, 4 | mp3an13 1448 | 1 ⊢ (𝐴 ∈ 𝑉 → ({〈∅, 𝐴〉, 〈1o, 𝐵〉}‘∅) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∅c0 4291 {cpr 4569 〈cop 4573 ‘cfv 6355 ωcom 7580 1oc1o 8095 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-res 5567 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fv 6363 df-om 7581 df-1o 8102 |
This theorem is referenced by: fvprif 16834 xpsfeq 16836 xpsfrnel2 16837 xpsff1o 16840 xpsle 16852 dmdprdpr 19171 dprdpr 19172 xpstopnlem1 22417 xpstopnlem2 22419 xpsxmetlem 22989 xpsdsval 22991 xpsmet 22992 |
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