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Mirrors > Home > ILE Home > Th. List > inimasn | 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 3300 | . . 3 ⊢ (𝑥 ∈ ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶})) ↔ (𝑥 ∈ (𝐴 “ {𝐶}) ∧ 𝑥 ∈ (𝐵 “ {𝐶}))) | |
2 | elin 3300 | . . . . 5 ⊢ (〈𝐶, 𝑥〉 ∈ (𝐴 ∩ 𝐵) ↔ (〈𝐶, 𝑥〉 ∈ 𝐴 ∧ 〈𝐶, 𝑥〉 ∈ 𝐵)) | |
3 | 2 | a1i 9 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → (〈𝐶, 𝑥〉 ∈ (𝐴 ∩ 𝐵) ↔ (〈𝐶, 𝑥〉 ∈ 𝐴 ∧ 〈𝐶, 𝑥〉 ∈ 𝐵))) |
4 | vex 2724 | . . . . 5 ⊢ 𝑥 ∈ V | |
5 | elimasng 4966 | . . . . 5 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ ((𝐴 ∩ 𝐵) “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ (𝐴 ∩ 𝐵))) | |
6 | 4, 5 | mpan2 422 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → (𝑥 ∈ ((𝐴 ∩ 𝐵) “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ (𝐴 ∩ 𝐵))) |
7 | elimasng 4966 | . . . . . 6 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐴 “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ 𝐴)) | |
8 | 4, 7 | mpan2 422 | . . . . 5 ⊢ (𝐶 ∈ 𝑉 → (𝑥 ∈ (𝐴 “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ 𝐴)) |
9 | elimasng 4966 | . . . . . 6 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐵 “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ 𝐵)) | |
10 | 4, 9 | mpan2 422 | . . . . 5 ⊢ (𝐶 ∈ 𝑉 → (𝑥 ∈ (𝐵 “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ 𝐵)) |
11 | 8, 10 | anbi12d 465 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → ((𝑥 ∈ (𝐴 “ {𝐶}) ∧ 𝑥 ∈ (𝐵 “ {𝐶})) ↔ (〈𝐶, 𝑥〉 ∈ 𝐴 ∧ 〈𝐶, 𝑥〉 ∈ 𝐵))) |
12 | 3, 6, 11 | 3bitr4rd 220 | . . 3 ⊢ (𝐶 ∈ 𝑉 → ((𝑥 ∈ (𝐴 “ {𝐶}) ∧ 𝑥 ∈ (𝐵 “ {𝐶})) ↔ 𝑥 ∈ ((𝐴 ∩ 𝐵) “ {𝐶}))) |
13 | 1, 12 | bitr2id 192 | . 2 ⊢ (𝐶 ∈ 𝑉 → (𝑥 ∈ ((𝐴 ∩ 𝐵) “ {𝐶}) ↔ 𝑥 ∈ ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶})))) |
14 | 13 | eqrdv 2162 | 1 ⊢ (𝐶 ∈ 𝑉 → ((𝐴 ∩ 𝐵) “ {𝐶}) = ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶}))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1342 ∈ wcel 2135 Vcvv 2721 ∩ cin 3110 {csn 3570 〈cop 3573 “ cima 4601 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-xp 4604 df-cnv 4606 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 |
This theorem is referenced by: (None) |
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