Theorem djur 7046

Description: A member of a disjoint union can be mapped from one of the classes which produced it. (Contributed by Jim Kingdon, 23-Jun-2022.) Upgrade implication to biconditional and shorten proof. (Revised by BJ, 14-Jul-2023.)

Assertion

Ref	Expression
djur	⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) ↔ (∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem djur

Step	Hyp	Ref	Expression
1		eldju 7045	. 2 ⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) ↔ (∃𝑥 ∈ 𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)))
2		fvres 5520	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝑥) = (inl‘𝑥))
3	2	eqeq2d 2182	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝐶 = ((inl ↾ 𝐴)‘𝑥) ↔ 𝐶 = (inl‘𝑥)))
4	3	rexbiia 2485	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ↔ ∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥))
5		fvres 5520	. . . . 5 ⊢ (𝑥 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝑥) = (inr‘𝑥))
6	5	eqeq2d 2182	. . . 4 ⊢ (𝑥 ∈ 𝐵 → (𝐶 = ((inr ↾ 𝐵)‘𝑥) ↔ 𝐶 = (inr‘𝑥)))
7	6	rexbiia 2485	. . 3 ⊢ (∃𝑥 ∈ 𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥) ↔ ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥))
8	4, 7	orbi12i 759	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)) ↔ (∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥)))
9	1, 8	bitri 183	1 ⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) ↔ (∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥)))

Syntax hints:   ↔ wb 104   ∨ wo 703   = wceq 1348   ∈ wcel 2141  ∃wrex 2449   ↾ cres 4613  ‘cfv 5198   ⊔ cdju 7014  inlcinl 7022  inrcinr 7023

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1st 6119  df-2nd 6120  df-1o 6395  df-dju 7015  df-inl 7024  df-inr 7025

This theorem is referenced by:  djuss  7047  updjud  7059  omp1eomlem  7071  0ct  7084  ctmlemr  7085  ctssdclemn0  7087  fodjuomnilemdc  7120  exmidfodomrlemeldju  7176