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Theorem mrcun 17566
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 1137 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋 ∧ 𝑉 βŠ† 𝑋) β†’ 𝐢 ∈ (Mooreβ€˜π‘‹))
2 mre1cl 17538 . . . . . . 7 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ 𝑋 ∈ 𝐢)
3 elpw2g 5345 . . . . . . 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 479 . . . . 5 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ π‘ˆ ∈ 𝒫 𝑋)
653adant3 1133 . . . 4 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋 ∧ 𝑉 βŠ† 𝑋) β†’ π‘ˆ ∈ 𝒫 𝑋)
7 elpw2g 5345 . . . . . . 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 479 . . . . 5 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑉 βŠ† 𝑋) β†’ 𝑉 ∈ 𝒫 𝑋)
1093adant2 1132 . . . 4 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋 ∧ 𝑉 βŠ† 𝑋) β†’ 𝑉 ∈ 𝒫 𝑋)
116, 10prssd 4826 . . 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 17565 . . 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 585 . 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 4926 . . . 4 ((π‘ˆ ∈ 𝒫 𝑋 ∧ 𝑉 ∈ 𝒫 𝑋) β†’ βˆͺ {π‘ˆ, 𝑉} = (π‘ˆ βˆͺ 𝑉))
166, 10, 15syl2anc 585 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋 ∧ 𝑉 βŠ† 𝑋) β†’ βˆͺ {π‘ˆ, 𝑉} = (π‘ˆ βˆͺ 𝑉))
1716fveq2d 6896 . 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 17553 . . . . . . . 8 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ 𝐹:𝒫 π‘‹βŸΆπΆ)
1918ffnd 6719 . . . . . . 7 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ 𝐹 Fn 𝒫 𝑋)
20193ad2ant1 1134 . . . . . 6 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋 ∧ 𝑉 βŠ† 𝑋) β†’ 𝐹 Fn 𝒫 𝑋)
21 fnimapr 6976 . . . . . 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 1372 . . . . 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 4923 . . . 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 6905 . . . . 5 (πΉβ€˜π‘ˆ) ∈ V
25 fvex 6905 . . . . 5 (πΉβ€˜π‘‰) ∈ V
2624, 25unipr 4927 . . . 4 βˆͺ {(πΉβ€˜π‘ˆ), (πΉβ€˜π‘‰)} = ((πΉβ€˜π‘ˆ) βˆͺ (πΉβ€˜π‘‰))
2723, 26eqtrdi 2789 . . 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 6896 . 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 2781 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 1088   = wceq 1542   ∈ wcel 2107   βˆͺ cun 3947   βŠ† wss 3949  π’« cpw 4603  {cpr 4631  βˆͺ cuni 4909   β€œ cima 5680   Fn wfn 6539  β€˜cfv 6544  Moorecmre 17526  mrClscmrc 17527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-mre 17530  df-mrc 17531
This theorem is referenced by: (None)
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