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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 7620 | . 2 ⊢ ∅ ∈ ω | |
2 | 1n0 8150 | . . 3 ⊢ 1o ≠ ∅ | |
3 | 2 | necomi 2988 | . 2 ⊢ ∅ ≠ 1o |
4 | fvpr1g 6964 | . 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 1542 ∈ wcel 2114 ≠ wne 2934 ∅c0 4211 {cpr 4518 〈cop 4522 ‘cfv 6339 ωcom 7599 1oc1o 8124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-res 5537 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fv 6347 df-om 7600 df-1o 8131 |
This theorem is referenced by: fvprif 16937 xpsfeq 16939 xpsfrnel2 16940 xpsff1o 16943 xpsle 16955 dmdprdpr 19290 dprdpr 19291 xpstopnlem1 22560 xpstopnlem2 22562 xpsxmetlem 23132 xpsdsval 23134 xpsmet 23135 |
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