ida2 - Metamath Proof Explorer

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Theorem ida2 18126

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 18125	. . 3 ⊢ (𝜑 → (𝐼‘𝑋) = ⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩)
7	6	fveq2d 6924	. 2 ⊢ (𝜑 → (2^nd ‘(𝐼‘𝑋)) = (2^nd ‘⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩))
8		fvex 6933	. . 3 ⊢ ( 1 ‘𝑋) ∈ V
9		ot3rdg 8046	. . 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 2796	1 ⊢ (𝜑 → (2^nd ‘(𝐼‘𝑋)) = ( 1 ‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 Vcvv 3488 ⟨cotp 4656 ‘cfv 6573 2^nd c2nd 8029 Basecbs 17258 Catccat 17722 Idccid 17723 Id_acida 18120

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-ot 4657 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-2nd 8031 df-ida 18122

This theorem is referenced by: arwlid 18139 arwrid 18140

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