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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 9208 | . . . . 5 ⊢ 1 ∈ ℤ | |
2 | fzpr 10002 | . . . . 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 8907 | . . . . 5 ⊢ 2 = (1 + 1) | |
5 | 4 | oveq2i 5847 | . . . 4 ⊢ (1...2) = (1...(1 + 1)) |
6 | 4 | preq2i 3651 | . . . 4 ⊢ {1, 2} = {1, (1 + 1)} |
7 | 3, 5, 6 | 3eqtr4i 2195 | . . 3 ⊢ (1...2) = {1, 2} |
8 | 7 | raleqi 2663 | . 2 ⊢ (∀𝑥 ∈ (1...2)(𝐹‘𝑥) = if(𝑥 = 1, 𝐴, 𝐵) ↔ ∀𝑥 ∈ {1, 2} (𝐹‘𝑥) = if(𝑥 = 1, 𝐴, 𝐵)) |
9 | 1ex 7885 | . . 3 ⊢ 1 ∈ V | |
10 | 2ex 8920 | . . 3 ⊢ 2 ∈ V | |
11 | fveq2 5480 | . . . 4 ⊢ (𝑥 = 1 → (𝐹‘𝑥) = (𝐹‘1)) | |
12 | iftrue 3520 | . . . 4 ⊢ (𝑥 = 1 → if(𝑥 = 1, 𝐴, 𝐵) = 𝐴) | |
13 | 11, 12 | eqeq12d 2179 | . . 3 ⊢ (𝑥 = 1 → ((𝐹‘𝑥) = if(𝑥 = 1, 𝐴, 𝐵) ↔ (𝐹‘1) = 𝐴)) |
14 | fveq2 5480 | . . . 4 ⊢ (𝑥 = 2 → (𝐹‘𝑥) = (𝐹‘2)) | |
15 | 1ne2 9054 | . . . . . . . 8 ⊢ 1 ≠ 2 | |
16 | 15 | necomi 2419 | . . . . . . 7 ⊢ 2 ≠ 1 |
17 | pm13.181 2416 | . . . . . . 7 ⊢ ((𝑥 = 2 ∧ 2 ≠ 1) → 𝑥 ≠ 1) | |
18 | 16, 17 | mpan2 422 | . . . . . 6 ⊢ (𝑥 = 2 → 𝑥 ≠ 1) |
19 | 18 | neneqd 2355 | . . . . 5 ⊢ (𝑥 = 2 → ¬ 𝑥 = 1) |
20 | 19 | iffalsed 3525 | . . . 4 ⊢ (𝑥 = 2 → if(𝑥 = 1, 𝐴, 𝐵) = 𝐵) |
21 | 14, 20 | eqeq12d 2179 | . . 3 ⊢ (𝑥 = 2 → ((𝐹‘𝑥) = if(𝑥 = 1, 𝐴, 𝐵) ↔ (𝐹‘2) = 𝐵)) |
22 | 9, 10, 13, 21 | ralpr 3625 | . 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 1342 ∈ wcel 2135 ≠ wne 2334 ∀wral 2442 ifcif 3515 {cpr 3571 ‘cfv 5182 (class class class)co 5836 1c1 7745 + caddc 7747 2c2 8899 ℤcz 9182 ...cfz 9935 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-2 8907 df-n0 9106 df-z 9183 df-uz 9458 df-fz 9936 |
This theorem is referenced by: (None) |
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