ida2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ida2		Structured version Visualization version GIF version

Theorem ida2 17974

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 17973	. . 3 ⊢ (𝜑 → (𝐼‘𝑋) = ⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩)
7	6	fveq2d 6835	. 2 ⊢ (𝜑 → (2^nd ‘(𝐼‘𝑋)) = (2^nd ‘⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩))
8		fvex 6844	. . 3 ⊢ ( 1 ‘𝑋) ∈ V
9		ot3rdg 7946	. . 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 2784	1 ⊢ (𝜑 → (2^nd ‘(𝐼‘𝑋)) = ( 1 ‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2113 Vcvv 3437 ⟨cotp 4585 ‘cfv 6489 2^nd c2nd 7929 Basecbs 17127 Catccat 17578 Idccid 17579 Id_acida 17968

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pr 5374 ax-un 7677

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-ral 3049 df-rex 3058 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4477 df-sn 4578 df-pr 4580 df-op 4584 df-ot 4586 df-uni 4861 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5516 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-2nd 7931 df-ida 17970

This theorem is referenced by: arwlid 17987 arwrid 17988

W3C validator