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Mirrors > Home > MPE Home > Th. List > idaval | Structured version Visualization version GIF version |
Description: Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
idafval.i | ⊢ 𝐼 = (Ida‘𝐶) |
idafval.b | ⊢ 𝐵 = (Base‘𝐶) |
idafval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
idafval.1 | ⊢ 1 = (Id‘𝐶) |
idaval.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
idaval | ⊢ (𝜑 → (𝐼‘𝑋) = 〈𝑋, 𝑋, ( 1 ‘𝑋)〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idafval.i | . . 3 ⊢ 𝐼 = (Ida‘𝐶) | |
2 | idafval.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
3 | idafval.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | idafval.1 | . . 3 ⊢ 1 = (Id‘𝐶) | |
5 | 1, 2, 3, 4 | idafval 17309 | . 2 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ 〈𝑥, 𝑥, ( 1 ‘𝑥)〉)) |
6 | simpr 488 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋) | |
7 | 6 | fveq2d 6649 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ( 1 ‘𝑥) = ( 1 ‘𝑋)) |
8 | 6, 6, 7 | oteq123d 4780 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 〈𝑥, 𝑥, ( 1 ‘𝑥)〉 = 〈𝑋, 𝑋, ( 1 ‘𝑋)〉) |
9 | idaval.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
10 | otex 5322 | . . 3 ⊢ 〈𝑋, 𝑋, ( 1 ‘𝑋)〉 ∈ V | |
11 | 10 | a1i 11 | . 2 ⊢ (𝜑 → 〈𝑋, 𝑋, ( 1 ‘𝑋)〉 ∈ V) |
12 | 5, 8, 9, 11 | fvmptd 6752 | 1 ⊢ (𝜑 → (𝐼‘𝑋) = 〈𝑋, 𝑋, ( 1 ‘𝑋)〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 〈cotp 4533 ‘cfv 6324 Basecbs 16475 Catccat 16927 Idccid 16928 Idacida 17305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-ot 4534 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ida 17307 |
This theorem is referenced by: ida2 17311 idahom 17312 |
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