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Mirrors > Home > MPE Home > Th. List > Mathboxes > elorvc | Structured version Visualization version GIF version |
Description: Elementhood of a preimage. (Contributed by Thierry Arnoux, 21-Jan-2017.) |
Ref | Expression |
---|---|
orvcval.1 | ⊢ (𝜑 → Fun 𝑋) |
orvcval.2 | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
orvcval.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑊) |
Ref | Expression |
---|---|
elorvc | ⊢ ((𝜑 ∧ 𝑧 ∈ dom 𝑋) → (𝑧 ∈ (𝑋∘RV/𝑐𝑅𝐴) ↔ (𝑋‘𝑧)𝑅𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orvcval.1 | . . . . 5 ⊢ (𝜑 → Fun 𝑋) | |
2 | orvcval.2 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
3 | orvcval.3 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑊) | |
4 | 1, 2, 3 | orvcval2 31718 | . . . 4 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧)𝑅𝐴}) |
5 | 4 | eleq2d 2900 | . . 3 ⊢ (𝜑 → (𝑧 ∈ (𝑋∘RV/𝑐𝑅𝐴) ↔ 𝑧 ∈ {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧)𝑅𝐴})) |
6 | rabid 3380 | . . 3 ⊢ (𝑧 ∈ {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧)𝑅𝐴} ↔ (𝑧 ∈ dom 𝑋 ∧ (𝑋‘𝑧)𝑅𝐴)) | |
7 | 5, 6 | syl6bb 289 | . 2 ⊢ (𝜑 → (𝑧 ∈ (𝑋∘RV/𝑐𝑅𝐴) ↔ (𝑧 ∈ dom 𝑋 ∧ (𝑋‘𝑧)𝑅𝐴))) |
8 | 7 | baibd 542 | 1 ⊢ ((𝜑 ∧ 𝑧 ∈ dom 𝑋) → (𝑧 ∈ (𝑋∘RV/𝑐𝑅𝐴) ↔ (𝑋‘𝑧)𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 {crab 3144 class class class wbr 5068 dom cdm 5557 Fun wfun 6351 ‘cfv 6357 (class class class)co 7158 ∘RV/𝑐corvc 31715 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-orvc 31716 |
This theorem is referenced by: elorrvc 31723 |
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