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Mathbox for Thierry Arnoux |
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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 34292 | . . . 4 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧)𝑅𝐴}) |
5 | 4 | eleq2d 2812 | . . 3 ⊢ (𝜑 → (𝑧 ∈ (𝑋∘RV/𝑐𝑅𝐴) ↔ 𝑧 ∈ {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧)𝑅𝐴})) |
6 | rabid 3440 | . . 3 ⊢ (𝑧 ∈ {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧)𝑅𝐴} ↔ (𝑧 ∈ dom 𝑋 ∧ (𝑋‘𝑧)𝑅𝐴)) | |
7 | 5, 6 | bitrdi 286 | . 2 ⊢ (𝜑 → (𝑧 ∈ (𝑋∘RV/𝑐𝑅𝐴) ↔ (𝑧 ∈ dom 𝑋 ∧ (𝑋‘𝑧)𝑅𝐴))) |
8 | 7 | baibd 538 | 1 ⊢ ((𝜑 ∧ 𝑧 ∈ dom 𝑋) → (𝑧 ∈ (𝑋∘RV/𝑐𝑅𝐴) ↔ (𝑋‘𝑧)𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2099 {crab 3419 class class class wbr 5153 dom cdm 5682 Fun wfun 6548 ‘cfv 6554 (class class class)co 7424 ∘RV/𝑐corvc 34289 |
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 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
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 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-orvc 34290 |
This theorem is referenced by: elorrvc 34297 |
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