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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 3225 | . . 3 ⊢ (𝑥 ∈ ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶})) ↔ (𝑥 ∈ (𝐴 “ {𝐶}) ∧ 𝑥 ∈ (𝐵 “ {𝐶}))) | |
2 | elin 3225 | . . . . 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 2660 | . . . . 5 ⊢ 𝑥 ∈ V | |
5 | elimasng 4865 | . . . . 5 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ ((𝐴 ∩ 𝐵) “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ (𝐴 ∩ 𝐵))) | |
6 | 4, 5 | mpan2 419 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → (𝑥 ∈ ((𝐴 ∩ 𝐵) “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ (𝐴 ∩ 𝐵))) |
7 | elimasng 4865 | . . . . . 6 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐴 “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ 𝐴)) | |
8 | 4, 7 | mpan2 419 | . . . . 5 ⊢ (𝐶 ∈ 𝑉 → (𝑥 ∈ (𝐴 “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ 𝐴)) |
9 | elimasng 4865 | . . . . . 6 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐵 “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ 𝐵)) | |
10 | 4, 9 | mpan2 419 | . . . . 5 ⊢ (𝐶 ∈ 𝑉 → (𝑥 ∈ (𝐵 “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ 𝐵)) |
11 | 8, 10 | anbi12d 462 | . . . 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 | syl5rbb 192 | . 2 ⊢ (𝐶 ∈ 𝑉 → (𝑥 ∈ ((𝐴 ∩ 𝐵) “ {𝐶}) ↔ 𝑥 ∈ ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶})))) |
14 | 13 | eqrdv 2113 | 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 1314 ∈ wcel 1463 Vcvv 2657 ∩ cin 3036 {csn 3493 〈cop 3496 “ cima 4502 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-v 2659 df-sbc 2879 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-br 3896 df-opab 3950 df-xp 4505 df-cnv 4507 df-dm 4509 df-rn 4510 df-res 4511 df-ima 4512 |
This theorem is referenced by: (None) |
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