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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 34461. (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.3 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑆MblFnM(sigaGen‘𝐽))) | |
| 2 | 1 | isanmbfm 34259 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ∪ ran MblFnM) | 
| 3 | 2 | mbfmfun 34255 | . . 3 ⊢ (𝜑 → Fun 𝑋) | 
| 4 | orvccel.1 | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) | |
| 5 | orvccel.2 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Top) | |
| 6 | 5 | sgsiga 34144 | . . . . . 6 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra) | 
| 7 | 4, 6, 1 | mbfmf 34256 | . . . . 5 ⊢ (𝜑 → 𝑋:∪ 𝑆⟶∪ (sigaGen‘𝐽)) | 
| 8 | elex 3500 | . . . . . . 7 ⊢ (𝐽 ∈ Top → 𝐽 ∈ V) | |
| 9 | unisg 34145 | . . . . . . 7 ⊢ (𝐽 ∈ V → ∪ (sigaGen‘𝐽) = ∪ 𝐽) | |
| 10 | 5, 8, 9 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → ∪ (sigaGen‘𝐽) = ∪ 𝐽) | 
| 11 | 10 | feq3d 6722 | . . . . 5 ⊢ (𝜑 → (𝑋:∪ 𝑆⟶∪ (sigaGen‘𝐽) ↔ 𝑋:∪ 𝑆⟶∪ 𝐽)) | 
| 12 | 7, 11 | mpbid 232 | . . . 4 ⊢ (𝜑 → 𝑋:∪ 𝑆⟶∪ 𝐽) | 
| 13 | 12 | frnd 6743 | . . 3 ⊢ (𝜑 → ran 𝑋 ⊆ ∪ 𝐽) | 
| 14 | fimacnvinrn2 7091 | . . 3 ⊢ ((Fun 𝑋 ∧ ran 𝑋 ⊆ ∪ 𝐽) → (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}) = (◡𝑋 “ ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽))) | |
| 15 | 3, 13, 14 | syl2anc 584 | . 2 ⊢ (𝜑 → (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}) = (◡𝑋 “ ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽))) | 
| 16 | orvccel.4 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 17 | 3, 1, 16 | orvcval 34461 | . 2 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴})) | 
| 18 | dfrab2 4319 | . . . 4 ⊢ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴} = ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽) | |
| 19 | 18 | a1i 11 | . . 3 ⊢ (𝜑 → {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴} = ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽)) | 
| 20 | 19 | imaeq2d 6077 | . 2 ⊢ (𝜑 → (◡𝑋 “ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴}) = (◡𝑋 “ ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽))) | 
| 21 | 15, 17, 20 | 3eqtr4d 2786 | 1 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴})) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 {cab 2713 {crab 3435 Vcvv 3479 ∩ cin 3949 ⊆ wss 3950 ∪ cuni 4906 class class class wbr 5142 ◡ccnv 5683 ran crn 5685 “ cima 5687 Fun wfun 6554 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 Topctop 22900 sigAlgebracsiga 34110 sigaGencsigagen 34140 MblFnMcmbfm 34251 ∘RV/𝑐corvc 34459 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fo 6566 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-map 8869 df-siga 34111 df-sigagen 34141 df-mbfm 34252 df-orvc 34460 | 
| This theorem is referenced by: orvcoel 34465 orvccel 34466 orrvcval4 34468 | 
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