1stinr - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > 1stinr		Structured version Visualization version GIF version

Theorem 1stinr 9358

Description: The first component of the value of a right injection is 1_o. (Contributed by AV, 27-Jun-2022.)

**Assertion**
Ref	Expression
1stinr	⊢ (𝑋 ∈ 𝑉 → (1^st ‘(inr‘𝑋)) = 1_o)

Proof of Theorem 1stinr Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-inr 9332	. . . 4 ⊢ inr = (𝑥 ∈ V ↦ ⟨1_o, 𝑥⟩)
2		opeq2 4804	. . . 4 ⊢ (𝑥 = 𝑋 → ⟨1_o, 𝑥⟩ = ⟨1_o, 𝑋⟩)
3		elex 3512	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
4		opex 5356	. . . . 5 ⊢ ⟨1_o, 𝑋⟩ ∈ V
5	4	a1i 11	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ⟨1_o, 𝑋⟩ ∈ V)
6	1, 2, 3, 5	fvmptd3 6791	. . 3 ⊢ (𝑋 ∈ 𝑉 → (inr‘𝑋) = ⟨1_o, 𝑋⟩)
7	6	fveq2d 6674	. 2 ⊢ (𝑋 ∈ 𝑉 → (1^st ‘(inr‘𝑋)) = (1^st ‘⟨1_o, 𝑋⟩))
8		1oex 8110	. . 3 ⊢ 1_o ∈ V
9		op1stg 7701	. . 3 ⊢ ((1_o ∈ V ∧ 𝑋 ∈ 𝑉) → (1^st ‘⟨1_o, 𝑋⟩) = 1_o)
10	8, 9	mpan 688	. 2 ⊢ (𝑋 ∈ 𝑉 → (1^st ‘⟨1_o, 𝑋⟩) = 1_o)
11	7, 10	eqtrd 2856	1 ⊢ (𝑋 ∈ 𝑉 → (1^st ‘(inr‘𝑋)) = 1_o)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ⟨cop 4573 ‘cfv 6355 1^st c1st 7687 1_oc1o 8095 inrcinr 9329

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-ord 6194 df-on 6195 df-suc 6197 df-iota 6314 df-fun 6357 df-fv 6363 df-1st 7689 df-1o 8102 df-inr 9332

This theorem is referenced by: updjudhcoinrg 9362