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Theorem eldju 7025
Description: Element of a disjoint union. (Contributed by BJ and Jim Kingdon, 23-Jun-2022.)
Assertion
Ref Expression
eldju (𝐶 ∈ (𝐴𝐵) ↔ (∃𝑥𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem eldju
StepHypRef Expression
1 djuunr 7023 . . . 4 (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (𝐴𝐵)
21eqcomi 2168 . . 3 (𝐴𝐵) = (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵))
32eleq2i 2231 . 2 (𝐶 ∈ (𝐴𝐵) ↔ 𝐶 ∈ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)))
4 elun 3259 . . 3 (𝐶 ∈ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) ↔ (𝐶 ∈ ran (inl ↾ 𝐴) ∨ 𝐶 ∈ ran (inr ↾ 𝐵)))
5 djulf1or 7013 . . . . . 6 (inl ↾ 𝐴):𝐴1-1-onto→({∅} × 𝐴)
6 f1ofn 5428 . . . . . 6 ((inl ↾ 𝐴):𝐴1-1-onto→({∅} × 𝐴) → (inl ↾ 𝐴) Fn 𝐴)
7 fvelrnb 5529 . . . . . 6 ((inl ↾ 𝐴) Fn 𝐴 → (𝐶 ∈ ran (inl ↾ 𝐴) ↔ ∃𝑥𝐴 ((inl ↾ 𝐴)‘𝑥) = 𝐶))
85, 6, 7mp2b 8 . . . . 5 (𝐶 ∈ ran (inl ↾ 𝐴) ↔ ∃𝑥𝐴 ((inl ↾ 𝐴)‘𝑥) = 𝐶)
9 eqcom 2166 . . . . . 6 (((inl ↾ 𝐴)‘𝑥) = 𝐶𝐶 = ((inl ↾ 𝐴)‘𝑥))
109rexbii 2471 . . . . 5 (∃𝑥𝐴 ((inl ↾ 𝐴)‘𝑥) = 𝐶 ↔ ∃𝑥𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥))
118, 10bitri 183 . . . 4 (𝐶 ∈ ran (inl ↾ 𝐴) ↔ ∃𝑥𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥))
12 djurf1or 7014 . . . . . 6 (inr ↾ 𝐵):𝐵1-1-onto→({1o} × 𝐵)
13 f1ofn 5428 . . . . . 6 ((inr ↾ 𝐵):𝐵1-1-onto→({1o} × 𝐵) → (inr ↾ 𝐵) Fn 𝐵)
14 fvelrnb 5529 . . . . . 6 ((inr ↾ 𝐵) Fn 𝐵 → (𝐶 ∈ ran (inr ↾ 𝐵) ↔ ∃𝑥𝐵 ((inr ↾ 𝐵)‘𝑥) = 𝐶))
1512, 13, 14mp2b 8 . . . . 5 (𝐶 ∈ ran (inr ↾ 𝐵) ↔ ∃𝑥𝐵 ((inr ↾ 𝐵)‘𝑥) = 𝐶)
16 eqcom 2166 . . . . . 6 (((inr ↾ 𝐵)‘𝑥) = 𝐶𝐶 = ((inr ↾ 𝐵)‘𝑥))
1716rexbii 2471 . . . . 5 (∃𝑥𝐵 ((inr ↾ 𝐵)‘𝑥) = 𝐶 ↔ ∃𝑥𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥))
1815, 17bitri 183 . . . 4 (𝐶 ∈ ran (inr ↾ 𝐵) ↔ ∃𝑥𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥))
1911, 18orbi12i 754 . . 3 ((𝐶 ∈ ran (inl ↾ 𝐴) ∨ 𝐶 ∈ ran (inr ↾ 𝐵)) ↔ (∃𝑥𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)))
204, 19bitri 183 . 2 (𝐶 ∈ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) ↔ (∃𝑥𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)))
213, 20bitri 183 1 (𝐶 ∈ (𝐴𝐵) ↔ (∃𝑥𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)))
Colors of variables: wff set class
Syntax hints:  wb 104  wo 698   = wceq 1342  wcel 2135  wrex 2443  cun 3110  c0 3405  {csn 3571   × cxp 4597  ran crn 4600  cres 4601   Fn wfn 5178  1-1-ontowf1o 5182  cfv 5183  1oc1o 6369  cdju 6994  inlcinl 7002  inrcinr 7003
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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-nul 4103  ax-pow 4148  ax-pr 4182  ax-un 4406
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2724  df-sbc 2948  df-dif 3114  df-un 3116  df-in 3118  df-ss 3125  df-nul 3406  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-uni 3785  df-br 3978  df-opab 4039  df-mpt 4040  df-tr 4076  df-id 4266  df-iord 4339  df-on 4341  df-suc 4344  df-xp 4605  df-rel 4606  df-cnv 4607  df-co 4608  df-dm 4609  df-rn 4610  df-res 4611  df-iota 5148  df-fun 5185  df-fn 5186  df-f 5187  df-f1 5188  df-fo 5189  df-f1o 5190  df-fv 5191  df-1st 6101  df-2nd 6102  df-1o 6376  df-dju 6995  df-inl 7004  df-inr 7005
This theorem is referenced by:  djur  7026  exmidfodomrlemreseldju  7148
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