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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 34439. (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 34238 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ∪ ran MblFnM) |
3 | 2 | mbfmfun 34234 | . . 3 ⊢ (𝜑 → Fun 𝑋) |
4 | orvccel.1 | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) | |
5 | orvccel.2 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Top) | |
6 | 5 | sgsiga 34123 | . . . . . 6 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra) |
7 | 4, 6, 1 | mbfmf 34235 | . . . . 5 ⊢ (𝜑 → 𝑋:∪ 𝑆⟶∪ (sigaGen‘𝐽)) |
8 | elex 3499 | . . . . . . 7 ⊢ (𝐽 ∈ Top → 𝐽 ∈ V) | |
9 | unisg 34124 | . . . . . . 7 ⊢ (𝐽 ∈ V → ∪ (sigaGen‘𝐽) = ∪ 𝐽) | |
10 | 5, 8, 9 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → ∪ (sigaGen‘𝐽) = ∪ 𝐽) |
11 | 10 | feq3d 6724 | . . . . 5 ⊢ (𝜑 → (𝑋:∪ 𝑆⟶∪ (sigaGen‘𝐽) ↔ 𝑋:∪ 𝑆⟶∪ 𝐽)) |
12 | 7, 11 | mpbid 232 | . . . 4 ⊢ (𝜑 → 𝑋:∪ 𝑆⟶∪ 𝐽) |
13 | 12 | frnd 6745 | . . 3 ⊢ (𝜑 → ran 𝑋 ⊆ ∪ 𝐽) |
14 | fimacnvinrn2 7092 | . . 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 34439 | . 2 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴})) |
18 | dfrab2 4326 | . . . 4 ⊢ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴} = ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽) | |
19 | 18 | a1i 11 | . . 3 ⊢ (𝜑 → {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴} = ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽)) |
20 | 19 | imaeq2d 6080 | . 2 ⊢ (𝜑 → (◡𝑋 “ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴}) = (◡𝑋 “ ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽))) |
21 | 15, 17, 20 | 3eqtr4d 2785 | 1 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {cab 2712 {crab 3433 Vcvv 3478 ∩ cin 3962 ⊆ wss 3963 ∪ cuni 4912 class class class wbr 5148 ◡ccnv 5688 ran crn 5690 “ cima 5692 Fun wfun 6557 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 Topctop 22915 sigAlgebracsiga 34089 sigaGencsigagen 34119 MblFnMcmbfm 34230 ∘RV/𝑐corvc 34437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-map 8867 df-siga 34090 df-sigagen 34120 df-mbfm 34231 df-orvc 34438 |
This theorem is referenced by: orvcoel 34443 orvccel 34444 orrvcval4 34446 |
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