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Theorem 1stinr 9350

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 9324	. . . 4 ⊢ inr = (𝑥 ∈ V ↦ ⟨1_o, 𝑥⟩)
2		opeq2 4796	. . . 4 ⊢ (𝑥 = 𝑋 → ⟨1_o, 𝑥⟩ = ⟨1_o, 𝑋⟩)
3		elex 3511	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
4		opex 5347	. . . . 5 ⊢ ⟨1_o, 𝑋⟩ ∈ V
5	4	a1i 11	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ⟨1_o, 𝑋⟩ ∈ V)
6	1, 2, 3, 5	fvmptd3 6784	. . 3 ⊢ (𝑋 ∈ 𝑉 → (inr‘𝑋) = ⟨1_o, 𝑋⟩)
7	6	fveq2d 6667	. 2 ⊢ (𝑋 ∈ 𝑉 → (1^st ‘(inr‘𝑋)) = (1^st ‘⟨1_o, 𝑋⟩))
8		1oex 8102	. . 3 ⊢ 1_o ∈ V
9		op1stg 7693	. . 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 2854	1 ⊢ (𝑋 ∈ 𝑉 → (1^st ‘(inr‘𝑋)) = 1_o)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1531 ∈ wcel 2108 Vcvv 3493 ⟨cop 4565 ‘cfv 6348 1^st c1st 7679 1_oc1o 8087 inrcinr 9321

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-ord 6187 df-on 6188 df-suc 6190 df-iota 6307 df-fun 6350 df-fv 6356 df-1st 7681 df-1o 8094 df-inr 9324

This theorem is referenced by: updjudhcoinrg 9354