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 31715 | . 2 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴})) |
5 | funfn 6385 | . . . 4 ⊢ (Fun 𝑋 ↔ 𝑋 Fn dom 𝑋) | |
6 | 1, 5 | sylib 220 | . . 3 ⊢ (𝜑 → 𝑋 Fn dom 𝑋) |
7 | fncnvima2 6831 | . . 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 6683 | . . . . 5 ⊢ (𝑋‘𝑧) ∈ V | |
10 | breq1 5069 | . . . . 5 ⊢ (𝑦 = (𝑋‘𝑧) → (𝑦𝑅𝐴 ↔ (𝑋‘𝑧)𝑅𝐴)) | |
11 | 9, 10 | elab 3667 | . . . 4 ⊢ ((𝑋‘𝑧) ∈ {𝑦 ∣ 𝑦𝑅𝐴} ↔ (𝑋‘𝑧)𝑅𝐴) |
12 | 11 | rabbii 3473 | . . 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 2860 | 1 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧)𝑅𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {cab 2799 {crab 3142 class class class wbr 5066 ◡ccnv 5554 dom cdm 5555 “ cima 5558 Fun wfun 6349 Fn wfn 6350 ‘cfv 6355 (class class class)co 7156 ∘RV/𝑐corvc 31713 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-orvc 31714 |
This theorem is referenced by: elorvc 31717 |
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