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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > orvcval4 | Structured version Visualization version GIF version |
Description: The value of the preimage mapping operator can be restricted to preimages in the base set of the topology. Cf. orvcval 31061. (Contributed by Thierry Arnoux, 21-Jan-2017.) |
Ref | Expression |
---|---|
orvccel.1 | ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) |
orvccel.2 | ⊢ (𝜑 → 𝐽 ∈ Top) |
orvccel.3 | ⊢ (𝜑 → 𝑋 ∈ (𝑆MblFnM(sigaGen‘𝐽))) |
orvccel.4 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
Ref | Expression |
---|---|
orvcval4 | ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orvccel.1 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) | |
2 | orvccel.2 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Top) | |
3 | 2 | sgsiga 30746 | . . . . 5 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra) |
4 | orvccel.3 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑆MblFnM(sigaGen‘𝐽))) | |
5 | 1, 3, 4 | isanmbfm 30859 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ∪ ran MblFnM) |
6 | 5 | mbfmfun 30857 | . . 3 ⊢ (𝜑 → Fun 𝑋) |
7 | 1, 3, 4 | mbfmf 30858 | . . . . 5 ⊢ (𝜑 → 𝑋:∪ 𝑆⟶∪ (sigaGen‘𝐽)) |
8 | elex 3429 | . . . . . . 7 ⊢ (𝐽 ∈ Top → 𝐽 ∈ V) | |
9 | unisg 30747 | . . . . . . 7 ⊢ (𝐽 ∈ V → ∪ (sigaGen‘𝐽) = ∪ 𝐽) | |
10 | 2, 8, 9 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → ∪ (sigaGen‘𝐽) = ∪ 𝐽) |
11 | 10 | feq3d 6269 | . . . . 5 ⊢ (𝜑 → (𝑋:∪ 𝑆⟶∪ (sigaGen‘𝐽) ↔ 𝑋:∪ 𝑆⟶∪ 𝐽)) |
12 | 7, 11 | mpbid 224 | . . . 4 ⊢ (𝜑 → 𝑋:∪ 𝑆⟶∪ 𝐽) |
13 | 12 | frnd 6289 | . . 3 ⊢ (𝜑 → ran 𝑋 ⊆ ∪ 𝐽) |
14 | fimacnvinrn2 6603 | . . 3 ⊢ ((Fun 𝑋 ∧ ran 𝑋 ⊆ ∪ 𝐽) → (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}) = (◡𝑋 “ ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽))) | |
15 | 6, 13, 14 | syl2anc 579 | . 2 ⊢ (𝜑 → (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}) = (◡𝑋 “ ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽))) |
16 | orvccel.4 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
17 | 6, 4, 16 | orvcval 31061 | . 2 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴})) |
18 | dfrab2 4134 | . . . 4 ⊢ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴} = ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽) | |
19 | 18 | a1i 11 | . . 3 ⊢ (𝜑 → {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴} = ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽)) |
20 | 19 | imaeq2d 5711 | . 2 ⊢ (𝜑 → (◡𝑋 “ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴}) = (◡𝑋 “ ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽))) |
21 | 15, 17, 20 | 3eqtr4d 2871 | 1 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 {cab 2811 {crab 3121 Vcvv 3414 ∩ cin 3797 ⊆ wss 3798 ∪ cuni 4660 class class class wbr 4875 ◡ccnv 5345 ran crn 5347 “ cima 5349 Fun wfun 6121 ⟶wf 6123 ‘cfv 6127 (class class class)co 6910 Topctop 21075 sigAlgebracsiga 30711 sigaGencsigagen 30742 MblFnMcmbfm 30853 ∘RV/𝑐corvc 31059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-fo 6133 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-1st 7433 df-2nd 7434 df-map 8129 df-siga 30712 df-sigagen 30743 df-mbfm 30854 df-orvc 31060 |
This theorem is referenced by: orvcoel 31065 orvccel 31066 orrvcval4 31068 |
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