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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > orvcval2 | Structured version Visualization version GIF version |
Description: Another way to express the value of the preimage mapping operator. (Contributed by Thierry Arnoux, 19-Jan-2017.) |
Ref | Expression |
---|---|
orvcval.1 | ⊢ (𝜑 → Fun 𝑋) |
orvcval.2 | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
orvcval.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑊) |
Ref | Expression |
---|---|
orvcval2 | ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧)𝑅𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orvcval.1 | . . 3 ⊢ (𝜑 → Fun 𝑋) | |
2 | orvcval.2 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
3 | orvcval.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑊) | |
4 | 1, 2, 3 | orvcval 34304 | . 2 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴})) |
5 | funfn 6581 | . . . 4 ⊢ (Fun 𝑋 ↔ 𝑋 Fn dom 𝑋) | |
6 | 1, 5 | sylib 217 | . . 3 ⊢ (𝜑 → 𝑋 Fn dom 𝑋) |
7 | fncnvima2 7066 | . . 3 ⊢ (𝑋 Fn dom 𝑋 → (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}) = {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧) ∈ {𝑦 ∣ 𝑦𝑅𝐴}}) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (𝜑 → (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}) = {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧) ∈ {𝑦 ∣ 𝑦𝑅𝐴}}) |
9 | fvex 6906 | . . . . 5 ⊢ (𝑋‘𝑧) ∈ V | |
10 | breq1 5148 | . . . . 5 ⊢ (𝑦 = (𝑋‘𝑧) → (𝑦𝑅𝐴 ↔ (𝑋‘𝑧)𝑅𝐴)) | |
11 | 9, 10 | elab 3665 | . . . 4 ⊢ ((𝑋‘𝑧) ∈ {𝑦 ∣ 𝑦𝑅𝐴} ↔ (𝑋‘𝑧)𝑅𝐴) |
12 | 11 | rabbii 3425 | . . 3 ⊢ {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧) ∈ {𝑦 ∣ 𝑦𝑅𝐴}} = {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧)𝑅𝐴} |
13 | 12 | a1i 11 | . 2 ⊢ (𝜑 → {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧) ∈ {𝑦 ∣ 𝑦𝑅𝐴}} = {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧)𝑅𝐴}) |
14 | 4, 8, 13 | 3eqtrd 2770 | 1 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧)𝑅𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 {cab 2703 {crab 3419 class class class wbr 5145 ◡ccnv 5673 dom cdm 5674 “ cima 5677 Fun wfun 6540 Fn wfn 6541 ‘cfv 6546 (class class class)co 7416 ∘RV/𝑐corvc 34302 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-fv 6554 df-ov 7419 df-oprab 7420 df-mpo 7421 df-orvc 34303 |
This theorem is referenced by: elorvc 34306 |
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