ida2 - Metamath Proof Explorer

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Theorem ida2 18246

Description: Morphism part of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)

**Hypotheses**
Ref	Expression
idafval.i	⊢ 𝐼 = (Id_a‘𝐶)
idafval.b	⊢ 𝐵 = (Base‘𝐶)
idafval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
idafval.1	⊢ 1 = (Id‘𝐶)
idaval.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

**Assertion**
Ref	Expression
ida2	⊢ (𝜑 → (2^nd ‘(𝐼‘𝑋)) = ( 1 ‘𝑋))

**Proof of Theorem ida2**
Step	Hyp	Ref	Expression
1		idafval.i	. . . 4 ⊢ 𝐼 = (Id_a‘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 18245	. . 3 ⊢ (𝜑 → (𝐼‘𝑋) = ⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩)
7	6	fveq2d 6897	. 2 ⊢ (𝜑 → (2^nd ‘(𝐼‘𝑋)) = (2^nd ‘⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩))
8		fvex 6906	. . 3 ⊢ ( 1 ‘𝑋) ∈ V
9		ot3rdg 8024	. . 3 ⊢ (( 1 ‘𝑋) ∈ V → (2^nd ‘⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩) = ( 1 ‘𝑋))
10	8, 9	ax-mp 5	. 2 ⊢ (2^nd ‘⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩) = ( 1 ‘𝑋)
11	7, 10	eqtrdi 2814	1 ⊢ (𝜑 → (2^nd ‘(𝐼‘𝑋)) = ( 1 ‘𝑋))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1569 ∈ wcel 2146 Vcvv 3453 ⟨cotp 4860 ‘cfv 6546 2^nd c2nd 8007 Basecbs 17399 Catccat 17850 Idccid 17851 Id_acida 18240

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5234 ax-sep 5251 ax-nul 5262 ax-pr 5394 ax-un 7758

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-reu 3369 df-rab 3416 df-v 3455 df-sbc 3742 df-csb 3851 df-ss 3900 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-sn 4588 df-pr 4590 df-op 4838 df-ot 4861 df-uni 4870 df-iun 4955 df-br 5106 df-opab 5170 df-mpt 5189 df-id 5549 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-2nd 8009 df-ida 18242

This theorem is used by: arwlid 18259 arwrid 18260

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