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Theorem mrcun 17531
Description: Idempotence of closure under a pair union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypothesis
Ref Expression
mrcfval.f 𝐹 = (mrClsβ€˜πΆ)
Assertion
Ref Expression
mrcun ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋 ∧ 𝑉 βŠ† 𝑋) β†’ (πΉβ€˜(π‘ˆ βˆͺ 𝑉)) = (πΉβ€˜((πΉβ€˜π‘ˆ) βˆͺ (πΉβ€˜π‘‰))))
Proof of Theorem mrcun
StepHypRef Expression
1 simp1 1136 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋 ∧ 𝑉 βŠ† 𝑋) β†’ 𝐢 ∈ (Mooreβ€˜π‘‹))
2 mre1cl 17503 . . . . . . 7 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ 𝑋 ∈ 𝐢)
3 elpw2g 5321 . . . . . . 7 (𝑋 ∈ 𝐢 β†’ (π‘ˆ ∈ 𝒫 𝑋 ↔ π‘ˆ βŠ† 𝑋))
42, 3syl 17 . . . . . 6 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ (π‘ˆ ∈ 𝒫 𝑋 ↔ π‘ˆ βŠ† 𝑋))
54biimpar 478 . . . . 5 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ π‘ˆ ∈ 𝒫 𝑋)
653adant3 1132 . . . 4 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋 ∧ 𝑉 βŠ† 𝑋) β†’ π‘ˆ ∈ 𝒫 𝑋)
7 elpw2g 5321 . . . . . . 7 (𝑋 ∈ 𝐢 β†’ (𝑉 ∈ 𝒫 𝑋 ↔ 𝑉 βŠ† 𝑋))
82, 7syl 17 . . . . . 6 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ (𝑉 ∈ 𝒫 𝑋 ↔ 𝑉 βŠ† 𝑋))
98biimpar 478 . . . . 5 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑉 βŠ† 𝑋) β†’ 𝑉 ∈ 𝒫 𝑋)
1093adant2 1131 . . . 4 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋 ∧ 𝑉 βŠ† 𝑋) β†’ 𝑉 ∈ 𝒫 𝑋)
116, 10prssd 4802 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋 ∧ 𝑉 βŠ† 𝑋) β†’ {π‘ˆ, 𝑉} βŠ† 𝒫 𝑋)
12 mrcfval.f . . . 4 𝐹 = (mrClsβ€˜πΆ)
1312mrcuni 17530 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ {π‘ˆ, 𝑉} βŠ† 𝒫 𝑋) β†’ (πΉβ€˜βˆͺ {π‘ˆ, 𝑉}) = (πΉβ€˜βˆͺ (𝐹 β€œ {π‘ˆ, 𝑉})))
141, 11, 13syl2anc 584 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋 ∧ 𝑉 βŠ† 𝑋) β†’ (πΉβ€˜βˆͺ {π‘ˆ, 𝑉}) = (πΉβ€˜βˆͺ (𝐹 β€œ {π‘ˆ, 𝑉})))
15 uniprg 4902 . . . 4 ((π‘ˆ ∈ 𝒫 𝑋 ∧ 𝑉 ∈ 𝒫 𝑋) β†’ βˆͺ {π‘ˆ, 𝑉} = (π‘ˆ βˆͺ 𝑉))
166, 10, 15syl2anc 584 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋 ∧ 𝑉 βŠ† 𝑋) β†’ βˆͺ {π‘ˆ, 𝑉} = (π‘ˆ βˆͺ 𝑉))
1716fveq2d 6866 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋 ∧ 𝑉 βŠ† 𝑋) β†’ (πΉβ€˜βˆͺ {π‘ˆ, 𝑉}) = (πΉβ€˜(π‘ˆ βˆͺ 𝑉)))
1812mrcf 17518 . . . . . . . 8 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ 𝐹:𝒫 π‘‹βŸΆπΆ)
1918ffnd 6689 . . . . . . 7 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ 𝐹 Fn 𝒫 𝑋)
20193ad2ant1 1133 . . . . . 6 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋 ∧ 𝑉 βŠ† 𝑋) β†’ 𝐹 Fn 𝒫 𝑋)
21 fnimapr 6945 . . . . . 6 ((𝐹 Fn 𝒫 𝑋 ∧ π‘ˆ ∈ 𝒫 𝑋 ∧ 𝑉 ∈ 𝒫 𝑋) β†’ (𝐹 β€œ {π‘ˆ, 𝑉}) = {(πΉβ€˜π‘ˆ), (πΉβ€˜π‘‰)})
2220, 6, 10, 21syl3anc 1371 . . . . 5 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋 ∧ 𝑉 βŠ† 𝑋) β†’ (𝐹 β€œ {π‘ˆ, 𝑉}) = {(πΉβ€˜π‘ˆ), (πΉβ€˜π‘‰)})
2322unieqd 4899 . . . 4 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋 ∧ 𝑉 βŠ† 𝑋) β†’ βˆͺ (𝐹 β€œ {π‘ˆ, 𝑉}) = βˆͺ {(πΉβ€˜π‘ˆ), (πΉβ€˜π‘‰)})
24 fvex 6875 . . . . 5 (πΉβ€˜π‘ˆ) ∈ V
25 fvex 6875 . . . . 5 (πΉβ€˜π‘‰) ∈ V
2624, 25unipr 4903 . . . 4 βˆͺ {(πΉβ€˜π‘ˆ), (πΉβ€˜π‘‰)} = ((πΉβ€˜π‘ˆ) βˆͺ (πΉβ€˜π‘‰))
2723, 26eqtrdi 2787 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋 ∧ 𝑉 βŠ† 𝑋) β†’ βˆͺ (𝐹 β€œ {π‘ˆ, 𝑉}) = ((πΉβ€˜π‘ˆ) βˆͺ (πΉβ€˜π‘‰)))
2827fveq2d 6866 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋 ∧ 𝑉 βŠ† 𝑋) β†’ (πΉβ€˜βˆͺ (𝐹 β€œ {π‘ˆ, 𝑉})) = (πΉβ€˜((πΉβ€˜π‘ˆ) βˆͺ (πΉβ€˜π‘‰))))
2914, 17, 283eqtr3d 2779 1 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋 ∧ 𝑉 βŠ† 𝑋) β†’ (πΉβ€˜(π‘ˆ βˆͺ 𝑉)) = (πΉβ€˜((πΉβ€˜π‘ˆ) βˆͺ (πΉβ€˜π‘‰))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   βˆͺ cun 3926   βŠ† wss 3928  π’« cpw 4580  {cpr 4608  βˆͺ cuni 4885   β€œ cima 5656   Fn wfn 6511  β€˜cfv 6516  Moorecmre 17491  mrClscmrc 17492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-int 4928  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-fv 6524  df-mre 17495  df-mrc 17496
This theorem is referenced by: (None)
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