ida2 - Metamath Proof Explorer

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Theorem ida2 18149

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 18148	. . 3 ⊢ (𝜑 → (𝐼‘𝑋) = ⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩)
7	6	fveq2d 6883	. 2 ⊢ (𝜑 → (2^nd ‘(𝐼‘𝑋)) = (2^nd ‘⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩))
8		fvex 6892	. . 3 ⊢ ( 1 ‘𝑋) ∈ V
9		ot3rdg 8003	. . 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 2811	1 ⊢ (𝜑 → (2^nd ‘(𝐼‘𝑋)) = ( 1 ‘𝑋))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2145 Vcvv 3450 ⟨cotp 4592 ‘cfv 6533 2^nd c2nd 7986 Basecbs 17302 Catccat 17753 Idccid 17754 Id_acida 18143

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pr 5398 ax-un 7737

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-ot 4593 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-2nd 7988 df-ida 18145

This theorem is used by: arwlid 18162 arwrid 18163

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