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Mirrors > Home > ILE Home > Th. List > fzprval | GIF version |
Description: Two ways of defining the first two values of a sequence on ℕ. (Contributed by NM, 5-Sep-2011.) |
Ref | Expression |
---|---|
fzprval | ⊢ (∀𝑥 ∈ (1...2)(𝐹‘𝑥) = if(𝑥 = 1, 𝐴, 𝐵) ↔ ((𝐹‘1) = 𝐴 ∧ (𝐹‘2) = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1z 9277 | . . . . 5 ⊢ 1 ∈ ℤ | |
2 | fzpr 10074 | . . . . 5 ⊢ (1 ∈ ℤ → (1...(1 + 1)) = {1, (1 + 1)}) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (1...(1 + 1)) = {1, (1 + 1)} |
4 | df-2 8976 | . . . . 5 ⊢ 2 = (1 + 1) | |
5 | 4 | oveq2i 5885 | . . . 4 ⊢ (1...2) = (1...(1 + 1)) |
6 | 4 | preq2i 3673 | . . . 4 ⊢ {1, 2} = {1, (1 + 1)} |
7 | 3, 5, 6 | 3eqtr4i 2208 | . . 3 ⊢ (1...2) = {1, 2} |
8 | 7 | raleqi 2676 | . 2 ⊢ (∀𝑥 ∈ (1...2)(𝐹‘𝑥) = if(𝑥 = 1, 𝐴, 𝐵) ↔ ∀𝑥 ∈ {1, 2} (𝐹‘𝑥) = if(𝑥 = 1, 𝐴, 𝐵)) |
9 | 1ex 7951 | . . 3 ⊢ 1 ∈ V | |
10 | 2ex 8989 | . . 3 ⊢ 2 ∈ V | |
11 | fveq2 5515 | . . . 4 ⊢ (𝑥 = 1 → (𝐹‘𝑥) = (𝐹‘1)) | |
12 | iftrue 3539 | . . . 4 ⊢ (𝑥 = 1 → if(𝑥 = 1, 𝐴, 𝐵) = 𝐴) | |
13 | 11, 12 | eqeq12d 2192 | . . 3 ⊢ (𝑥 = 1 → ((𝐹‘𝑥) = if(𝑥 = 1, 𝐴, 𝐵) ↔ (𝐹‘1) = 𝐴)) |
14 | fveq2 5515 | . . . 4 ⊢ (𝑥 = 2 → (𝐹‘𝑥) = (𝐹‘2)) | |
15 | 1ne2 9123 | . . . . . . . 8 ⊢ 1 ≠ 2 | |
16 | 15 | necomi 2432 | . . . . . . 7 ⊢ 2 ≠ 1 |
17 | pm13.181 2429 | . . . . . . 7 ⊢ ((𝑥 = 2 ∧ 2 ≠ 1) → 𝑥 ≠ 1) | |
18 | 16, 17 | mpan2 425 | . . . . . 6 ⊢ (𝑥 = 2 → 𝑥 ≠ 1) |
19 | 18 | neneqd 2368 | . . . . 5 ⊢ (𝑥 = 2 → ¬ 𝑥 = 1) |
20 | 19 | iffalsed 3544 | . . . 4 ⊢ (𝑥 = 2 → if(𝑥 = 1, 𝐴, 𝐵) = 𝐵) |
21 | 14, 20 | eqeq12d 2192 | . . 3 ⊢ (𝑥 = 2 → ((𝐹‘𝑥) = if(𝑥 = 1, 𝐴, 𝐵) ↔ (𝐹‘2) = 𝐵)) |
22 | 9, 10, 13, 21 | ralpr 3647 | . 2 ⊢ (∀𝑥 ∈ {1, 2} (𝐹‘𝑥) = if(𝑥 = 1, 𝐴, 𝐵) ↔ ((𝐹‘1) = 𝐴 ∧ (𝐹‘2) = 𝐵)) |
23 | 8, 22 | bitri 184 | 1 ⊢ (∀𝑥 ∈ (1...2)(𝐹‘𝑥) = if(𝑥 = 1, 𝐴, 𝐵) ↔ ((𝐹‘1) = 𝐴 ∧ (𝐹‘2) = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 ∀wral 2455 ifcif 3534 {cpr 3593 ‘cfv 5216 (class class class)co 5874 1c1 7811 + caddc 7813 2c2 8968 ℤcz 9251 ...cfz 10006 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-inn 8918 df-2 8976 df-n0 9175 df-z 9252 df-uz 9527 df-fz 10007 |
This theorem is referenced by: (None) |
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