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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 18017 | . 2 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ ⟨𝑥, 𝑥, ( 1 ‘𝑥)⟩)) |
6 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋) | |
7 | 6 | fveq2d 6888 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ( 1 ‘𝑥) = ( 1 ‘𝑋)) |
8 | 6, 6, 7 | oteq123d 4883 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ⟨𝑥, 𝑥, ( 1 ‘𝑥)⟩ = ⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩) |
9 | idaval.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
10 | otex 5458 | . . 3 ⊢ ⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩ ∈ V | |
11 | 10 | a1i 11 | . 2 ⊢ (𝜑 → ⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩ ∈ V) |
12 | 5, 8, 9, 11 | fvmptd 6998 | 1 ⊢ (𝜑 → (𝐼‘𝑋) = ⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ⟨cotp 4631 ‘cfv 6536 Basecbs 17151 Catccat 17615 Idccid 17616 Idacida 18013 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-ot 4632 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ida 18015 |
This theorem is referenced by: ida2 18019 idahom 18020 |
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