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Theorem joindef 18338
Description: Two ways to say that a join is defined. (Contributed by NM, 9-Sep-2018.)
Hypotheses
Ref Expression
joindef.u π‘ˆ = (lubβ€˜πΎ)
joindef.j ∨ = (joinβ€˜πΎ)
joindef.k (πœ‘ β†’ 𝐾 ∈ 𝑉)
joindef.x (πœ‘ β†’ 𝑋 ∈ π‘Š)
joindef.y (πœ‘ β†’ π‘Œ ∈ 𝑍)
Assertion
Ref Expression
joindef (πœ‘ β†’ (βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∨ ↔ {𝑋, π‘Œ} ∈ dom π‘ˆ))

Proof of Theorem joindef
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 joindef.k . . 3 (πœ‘ β†’ 𝐾 ∈ 𝑉)
2 joindef.u . . . . 5 π‘ˆ = (lubβ€˜πΎ)
3 joindef.j . . . . 5 ∨ = (joinβ€˜πΎ)
42, 3joindm 18337 . . . 4 (𝐾 ∈ 𝑉 β†’ dom ∨ = {⟨π‘₯, π‘¦βŸ© ∣ {π‘₯, 𝑦} ∈ dom π‘ˆ})
54eleq2d 2813 . . 3 (𝐾 ∈ 𝑉 β†’ (βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∨ ↔ βŸ¨π‘‹, π‘ŒβŸ© ∈ {⟨π‘₯, π‘¦βŸ© ∣ {π‘₯, 𝑦} ∈ dom π‘ˆ}))
61, 5syl 17 . 2 (πœ‘ β†’ (βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∨ ↔ βŸ¨π‘‹, π‘ŒβŸ© ∈ {⟨π‘₯, π‘¦βŸ© ∣ {π‘₯, 𝑦} ∈ dom π‘ˆ}))
7 joindef.x . . 3 (πœ‘ β†’ 𝑋 ∈ π‘Š)
8 joindef.y . . 3 (πœ‘ β†’ π‘Œ ∈ 𝑍)
9 preq1 4732 . . . . 5 (π‘₯ = 𝑋 β†’ {π‘₯, 𝑦} = {𝑋, 𝑦})
109eleq1d 2812 . . . 4 (π‘₯ = 𝑋 β†’ ({π‘₯, 𝑦} ∈ dom π‘ˆ ↔ {𝑋, 𝑦} ∈ dom π‘ˆ))
11 preq2 4733 . . . . 5 (𝑦 = π‘Œ β†’ {𝑋, 𝑦} = {𝑋, π‘Œ})
1211eleq1d 2812 . . . 4 (𝑦 = π‘Œ β†’ ({𝑋, 𝑦} ∈ dom π‘ˆ ↔ {𝑋, π‘Œ} ∈ dom π‘ˆ))
1310, 12opelopabg 5531 . . 3 ((𝑋 ∈ π‘Š ∧ π‘Œ ∈ 𝑍) β†’ (βŸ¨π‘‹, π‘ŒβŸ© ∈ {⟨π‘₯, π‘¦βŸ© ∣ {π‘₯, 𝑦} ∈ dom π‘ˆ} ↔ {𝑋, π‘Œ} ∈ dom π‘ˆ))
147, 8, 13syl2anc 583 . 2 (πœ‘ β†’ (βŸ¨π‘‹, π‘ŒβŸ© ∈ {⟨π‘₯, π‘¦βŸ© ∣ {π‘₯, 𝑦} ∈ dom π‘ˆ} ↔ {𝑋, π‘Œ} ∈ dom π‘ˆ))
156, 14bitrd 279 1 (πœ‘ β†’ (βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∨ ↔ {𝑋, π‘Œ} ∈ dom π‘ˆ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  {cpr 4625  βŸ¨cop 4629  {copab 5203  dom cdm 5669  β€˜cfv 6536  lubclub 18271  joincjn 18273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-oprab 7408  df-lub 18308  df-join 18310
This theorem is referenced by:  joinval  18339  joincl  18340  joindmss  18341  joineu  18344  clatl  18470  joindm2  47857
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