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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 18102 | . 2 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ 〈𝑥, 𝑥, ( 1 ‘𝑥)〉)) | 
| 6 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋) | |
| 7 | 6 | fveq2d 6910 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ( 1 ‘𝑥) = ( 1 ‘𝑋)) | 
| 8 | 6, 6, 7 | oteq123d 4888 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 〈𝑥, 𝑥, ( 1 ‘𝑥)〉 = 〈𝑋, 𝑋, ( 1 ‘𝑋)〉) | 
| 9 | idaval.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 10 | otex 5470 | . . 3 ⊢ 〈𝑋, 𝑋, ( 1 ‘𝑋)〉 ∈ V | |
| 11 | 10 | a1i 11 | . 2 ⊢ (𝜑 → 〈𝑋, 𝑋, ( 1 ‘𝑋)〉 ∈ V) | 
| 12 | 5, 8, 9, 11 | fvmptd 7023 | 1 ⊢ (𝜑 → (𝐼‘𝑋) = 〈𝑋, 𝑋, ( 1 ‘𝑋)〉) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cotp 4634 ‘cfv 6561 Basecbs 17247 Catccat 17707 Idccid 17708 Idacida 18098 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-ot 4635 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ida 18100 | 
| This theorem is referenced by: ida2 18104 idahom 18105 | 
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