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Mirrors > Home > MPE Home > Th. List > inimasn | Structured version Visualization version GIF version |
Description: The intersection of the image of singleton. (Contributed by Thierry Arnoux, 16-Dec-2017.) |
Ref | Expression |
---|---|
inimasn | ⊢ (𝐶 ∈ 𝑉 → ((𝐴 ∩ 𝐵) “ {𝐶}) = ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 4023 | . . 3 ⊢ (𝑥 ∈ ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶})) ↔ (𝑥 ∈ (𝐴 “ {𝐶}) ∧ 𝑥 ∈ (𝐵 “ {𝐶}))) | |
2 | elin 4023 | . . . . 5 ⊢ (〈𝐶, 𝑥〉 ∈ (𝐴 ∩ 𝐵) ↔ (〈𝐶, 𝑥〉 ∈ 𝐴 ∧ 〈𝐶, 𝑥〉 ∈ 𝐵)) | |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → (〈𝐶, 𝑥〉 ∈ (𝐴 ∩ 𝐵) ↔ (〈𝐶, 𝑥〉 ∈ 𝐴 ∧ 〈𝐶, 𝑥〉 ∈ 𝐵))) |
4 | elimasng 5732 | . . . . 5 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ ((𝐴 ∩ 𝐵) “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ (𝐴 ∩ 𝐵))) | |
5 | 4 | elvd 3419 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → (𝑥 ∈ ((𝐴 ∩ 𝐵) “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ (𝐴 ∩ 𝐵))) |
6 | elimasng 5732 | . . . . . 6 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐴 “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ 𝐴)) | |
7 | 6 | elvd 3419 | . . . . 5 ⊢ (𝐶 ∈ 𝑉 → (𝑥 ∈ (𝐴 “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ 𝐴)) |
8 | elimasng 5732 | . . . . . 6 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐵 “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ 𝐵)) | |
9 | 8 | elvd 3419 | . . . . 5 ⊢ (𝐶 ∈ 𝑉 → (𝑥 ∈ (𝐵 “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ 𝐵)) |
10 | 7, 9 | anbi12d 624 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → ((𝑥 ∈ (𝐴 “ {𝐶}) ∧ 𝑥 ∈ (𝐵 “ {𝐶})) ↔ (〈𝐶, 𝑥〉 ∈ 𝐴 ∧ 〈𝐶, 𝑥〉 ∈ 𝐵))) |
11 | 3, 5, 10 | 3bitr4rd 304 | . . 3 ⊢ (𝐶 ∈ 𝑉 → ((𝑥 ∈ (𝐴 “ {𝐶}) ∧ 𝑥 ∈ (𝐵 “ {𝐶})) ↔ 𝑥 ∈ ((𝐴 ∩ 𝐵) “ {𝐶}))) |
12 | 1, 11 | syl5rbb 276 | . 2 ⊢ (𝐶 ∈ 𝑉 → (𝑥 ∈ ((𝐴 ∩ 𝐵) “ {𝐶}) ↔ 𝑥 ∈ ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶})))) |
13 | 12 | eqrdv 2823 | 1 ⊢ (𝐶 ∈ 𝑉 → ((𝐴 ∩ 𝐵) “ {𝐶}) = ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1656 ∈ wcel 2164 Vcvv 3414 ∩ cin 3797 {csn 4397 〈cop 4403 “ cima 5345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-br 4874 df-opab 4936 df-xp 5348 df-cnv 5350 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 |
This theorem is referenced by: restutopopn 22412 ustuqtop2 22416 |
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