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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 17147 | . . 3 ⊢ (𝜑 → (𝐼‘𝑋) = 〈𝑋, 𝑋, ( 1 ‘𝑋)〉) |
7 | 6 | fveq2d 6542 | . 2 ⊢ (𝜑 → (2nd ‘(𝐼‘𝑋)) = (2nd ‘〈𝑋, 𝑋, ( 1 ‘𝑋)〉)) |
8 | fvex 6551 | . . 3 ⊢ ( 1 ‘𝑋) ∈ V | |
9 | ot3rdg 7561 | . . 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 | syl6eq 2847 | 1 ⊢ (𝜑 → (2nd ‘(𝐼‘𝑋)) = ( 1 ‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1522 ∈ wcel 2081 Vcvv 3437 〈cotp 4480 ‘cfv 6225 2nd c2nd 7544 Basecbs 16312 Catccat 16764 Idccid 16765 Idacida 17142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-ot 4481 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-2nd 7546 df-ida 17144 |
This theorem is referenced by: arwlid 17161 arwrid 17162 |
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