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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 9080 | . . . . 5 ⊢ 1 ∈ ℤ | |
2 | fzpr 9857 | . . . . 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 8779 | . . . . 5 ⊢ 2 = (1 + 1) | |
5 | 4 | oveq2i 5785 | . . . 4 ⊢ (1...2) = (1...(1 + 1)) |
6 | 4 | preq2i 3604 | . . . 4 ⊢ {1, 2} = {1, (1 + 1)} |
7 | 3, 5, 6 | 3eqtr4i 2170 | . . 3 ⊢ (1...2) = {1, 2} |
8 | 7 | raleqi 2630 | . 2 ⊢ (∀𝑥 ∈ (1...2)(𝐹‘𝑥) = if(𝑥 = 1, 𝐴, 𝐵) ↔ ∀𝑥 ∈ {1, 2} (𝐹‘𝑥) = if(𝑥 = 1, 𝐴, 𝐵)) |
9 | 1ex 7761 | . . 3 ⊢ 1 ∈ V | |
10 | 2ex 8792 | . . 3 ⊢ 2 ∈ V | |
11 | fveq2 5421 | . . . 4 ⊢ (𝑥 = 1 → (𝐹‘𝑥) = (𝐹‘1)) | |
12 | iftrue 3479 | . . . 4 ⊢ (𝑥 = 1 → if(𝑥 = 1, 𝐴, 𝐵) = 𝐴) | |
13 | 11, 12 | eqeq12d 2154 | . . 3 ⊢ (𝑥 = 1 → ((𝐹‘𝑥) = if(𝑥 = 1, 𝐴, 𝐵) ↔ (𝐹‘1) = 𝐴)) |
14 | fveq2 5421 | . . . 4 ⊢ (𝑥 = 2 → (𝐹‘𝑥) = (𝐹‘2)) | |
15 | 1ne2 8926 | . . . . . . . 8 ⊢ 1 ≠ 2 | |
16 | 15 | necomi 2393 | . . . . . . 7 ⊢ 2 ≠ 1 |
17 | pm13.181 2390 | . . . . . . 7 ⊢ ((𝑥 = 2 ∧ 2 ≠ 1) → 𝑥 ≠ 1) | |
18 | 16, 17 | mpan2 421 | . . . . . 6 ⊢ (𝑥 = 2 → 𝑥 ≠ 1) |
19 | 18 | neneqd 2329 | . . . . 5 ⊢ (𝑥 = 2 → ¬ 𝑥 = 1) |
20 | 19 | iffalsed 3484 | . . . 4 ⊢ (𝑥 = 2 → if(𝑥 = 1, 𝐴, 𝐵) = 𝐵) |
21 | 14, 20 | eqeq12d 2154 | . . 3 ⊢ (𝑥 = 2 → ((𝐹‘𝑥) = if(𝑥 = 1, 𝐴, 𝐵) ↔ (𝐹‘2) = 𝐵)) |
22 | 9, 10, 13, 21 | ralpr 3578 | . 2 ⊢ (∀𝑥 ∈ {1, 2} (𝐹‘𝑥) = if(𝑥 = 1, 𝐴, 𝐵) ↔ ((𝐹‘1) = 𝐴 ∧ (𝐹‘2) = 𝐵)) |
23 | 8, 22 | bitri 183 | 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 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ≠ wne 2308 ∀wral 2416 ifcif 3474 {cpr 3528 ‘cfv 5123 (class class class)co 5774 1c1 7621 + caddc 7623 2c2 8771 ℤcz 9054 ...cfz 9790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-2 8779 df-n0 8978 df-z 9055 df-uz 9327 df-fz 9791 |
This theorem is referenced by: (None) |
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