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Mirrors > Home > MPE Home > Th. List > ida2 | Structured version Visualization version GIF version |
Description: Morphism part of the identity arrow. (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 |
---|---|
ida2 | ⊢ (𝜑 → (2nd ‘(𝐼‘𝑋)) = ( 1 ‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idafval.i | . . . 4 ⊢ 𝐼 = (Ida‘𝐶) | |
2 | idafval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
3 | idafval.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | idafval.1 | . . . 4 ⊢ 1 = (Id‘𝐶) | |
5 | idaval.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | 1, 2, 3, 4, 5 | idaval 18040 | . . 3 ⊢ (𝜑 → (𝐼‘𝑋) = ⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩) |
7 | 6 | fveq2d 6895 | . 2 ⊢ (𝜑 → (2nd ‘(𝐼‘𝑋)) = (2nd ‘⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩)) |
8 | fvex 6904 | . . 3 ⊢ ( 1 ‘𝑋) ∈ V | |
9 | ot3rdg 8003 | . . 3 ⊢ (( 1 ‘𝑋) ∈ V → (2nd ‘⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩) = ( 1 ‘𝑋)) | |
10 | 8, 9 | ax-mp 5 | . 2 ⊢ (2nd ‘⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩) = ( 1 ‘𝑋) |
11 | 7, 10 | eqtrdi 2784 | 1 ⊢ (𝜑 → (2nd ‘(𝐼‘𝑋)) = ( 1 ‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3470 ⟨cotp 4632 ‘cfv 6542 2nd c2nd 7986 Basecbs 17173 Catccat 17637 Idccid 17638 Idacida 18035 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-ot 4633 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-2nd 7988 df-ida 18037 |
This theorem is referenced by: arwlid 18054 arwrid 18055 |
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