Theorem inrresf 9339

Description: The right injection restricted to the right class of a disjoint union is a function from the right class into the disjoint union. (Contributed by AV, 27-Jun-2022.)

Assertion

Ref	Expression
inrresf	⊢ (inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵)

Proof of Theorem inrresf

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		djurf1o 9336	. . 3 ⊢ inr:V–1-1-onto→({1o} × V)
2		f1ofun 6612	. . 3 ⊢ (inr:V–1-1-onto→({1o} × V) → Fun inr)
3		ffvresb 6883	. . 3 ⊢ (Fun inr → ((inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ dom inr ∧ (inr‘𝑥) ∈ (𝐴 ⊔ 𝐵))))
4	1, 2, 3	mp2b 10	. 2 ⊢ ((inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ dom inr ∧ (inr‘𝑥) ∈ (𝐴 ⊔ 𝐵)))
5		elex 3513	. . . 4 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ V)
6		opex 5349	. . . . 5 ⊢ ⟨1o, 𝑥⟩ ∈ V
7		df-inr 9326	. . . . 5 ⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
8	6, 7	dmmpti 6487	. . . 4 ⊢ dom inr = V
9	5, 8	eleqtrrdi 2924	. . 3 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ dom inr)
10		djurcl 9334	. . 3 ⊢ (𝑥 ∈ 𝐵 → (inr‘𝑥) ∈ (𝐴 ⊔ 𝐵))
11	9, 10	jca 514	. 2 ⊢ (𝑥 ∈ 𝐵 → (𝑥 ∈ dom inr ∧ (inr‘𝑥) ∈ (𝐴 ⊔ 𝐵)))
12	4, 11	mprgbir 3153	1 ⊢ (inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵)

Syntax hints:   ↔ wb 208   ∧ wa 398   ∈ wcel 2110  ∀wral 3138  Vcvv 3495  {csn 4561  ⟨cop 4567   × cxp 5548  dom cdm 5550   ↾ cres 5552  Fun wfun 6344  ⟶wf 6346  –1-1-onto→wf1o 6349  ‘cfv 6350  1_oc1o 8089   ⊔ cdju 9321  inrcinr 9323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  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 3497  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 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-om 7575  df-1st 7683  df-2nd 7684  df-1o 8096  df-dju 9324  df-inr 9326

This theorem is referenced by:  inrresf1  9340  updjudhcoinrg  9356  updjud  9357