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Mirrors > Home > MPE Home > Th. List > xpsfeq | Structured version Visualization version GIF version |
Description: A function on 2o is determined by its values at zero and one. (Contributed by Mario Carneiro, 27-Aug-2015.) |
Ref | Expression |
---|---|
xpsfeq | ⊢ (𝐺 Fn 2o → {⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩} = 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6856 | . . . 4 ⊢ (𝐺‘∅) ∈ V | |
2 | fvex 6856 | . . . 4 ⊢ (𝐺‘1o) ∈ V | |
3 | fnpr2o 17440 | . . . 4 ⊢ (((𝐺‘∅) ∈ V ∧ (𝐺‘1o) ∈ V) → {⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩} Fn 2o) | |
4 | 1, 2, 3 | mp2an 691 | . . 3 ⊢ {⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩} Fn 2o |
5 | 4 | a1i 11 | . 2 ⊢ (𝐺 Fn 2o → {⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩} Fn 2o) |
6 | id 22 | . 2 ⊢ (𝐺 Fn 2o → 𝐺 Fn 2o) | |
7 | elpri 4609 | . . . . 5 ⊢ (𝑘 ∈ {∅, 1o} → (𝑘 = ∅ ∨ 𝑘 = 1o)) | |
8 | df2o3 8421 | . . . . 5 ⊢ 2o = {∅, 1o} | |
9 | 7, 8 | eleq2s 2856 | . . . 4 ⊢ (𝑘 ∈ 2o → (𝑘 = ∅ ∨ 𝑘 = 1o)) |
10 | fvpr0o 17442 | . . . . . . 7 ⊢ ((𝐺‘∅) ∈ V → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘∅) = (𝐺‘∅)) | |
11 | 1, 10 | ax-mp 5 | . . . . . 6 ⊢ ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘∅) = (𝐺‘∅) |
12 | fveq2 6843 | . . . . . 6 ⊢ (𝑘 = ∅ → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘∅)) | |
13 | fveq2 6843 | . . . . . 6 ⊢ (𝑘 = ∅ → (𝐺‘𝑘) = (𝐺‘∅)) | |
14 | 11, 12, 13 | 3eqtr4a 2803 | . . . . 5 ⊢ (𝑘 = ∅ → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺‘𝑘)) |
15 | fvpr1o 17443 | . . . . . . 7 ⊢ ((𝐺‘1o) ∈ V → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘1o) = (𝐺‘1o)) | |
16 | 2, 15 | ax-mp 5 | . . . . . 6 ⊢ ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘1o) = (𝐺‘1o) |
17 | fveq2 6843 | . . . . . 6 ⊢ (𝑘 = 1o → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘1o)) | |
18 | fveq2 6843 | . . . . . 6 ⊢ (𝑘 = 1o → (𝐺‘𝑘) = (𝐺‘1o)) | |
19 | 16, 17, 18 | 3eqtr4a 2803 | . . . . 5 ⊢ (𝑘 = 1o → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺‘𝑘)) |
20 | 14, 19 | jaoi 856 | . . . 4 ⊢ ((𝑘 = ∅ ∨ 𝑘 = 1o) → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺‘𝑘)) |
21 | 9, 20 | syl 17 | . . 3 ⊢ (𝑘 ∈ 2o → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺‘𝑘)) |
22 | 21 | adantl 483 | . 2 ⊢ ((𝐺 Fn 2o ∧ 𝑘 ∈ 2o) → ({⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩}‘𝑘) = (𝐺‘𝑘)) |
23 | 5, 6, 22 | eqfnfvd 6986 | 1 ⊢ (𝐺 Fn 2o → {⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩} = 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 846 = wceq 1542 ∈ wcel 2107 Vcvv 3446 ∅c0 4283 {cpr 4589 ⟨cop 4593 Fn wfn 6492 ‘cfv 6497 1oc1o 8406 2oc2o 8407 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-fv 6505 df-om 7804 df-1o 8413 df-2o 8414 |
This theorem is referenced by: xpsff1o 17450 xpstopnlem2 23165 |
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