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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 4166 | . . 3 ⊢ (𝑥 ∈ ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶})) ↔ (𝑥 ∈ (𝐴 “ {𝐶}) ∧ 𝑥 ∈ (𝐵 “ {𝐶}))) | |
2 | elin 4166 | . . . . 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 5948 | . . . . 5 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ ((𝐴 ∩ 𝐵) “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ (𝐴 ∩ 𝐵))) | |
5 | 4 | elvd 3498 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → (𝑥 ∈ ((𝐴 ∩ 𝐵) “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ (𝐴 ∩ 𝐵))) |
6 | elimasng 5948 | . . . . . 6 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐴 “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ 𝐴)) | |
7 | 6 | elvd 3498 | . . . . 5 ⊢ (𝐶 ∈ 𝑉 → (𝑥 ∈ (𝐴 “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ 𝐴)) |
8 | elimasng 5948 | . . . . . 6 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐵 “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ 𝐵)) | |
9 | 8 | elvd 3498 | . . . . 5 ⊢ (𝐶 ∈ 𝑉 → (𝑥 ∈ (𝐵 “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ 𝐵)) |
10 | 7, 9 | anbi12d 630 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → ((𝑥 ∈ (𝐴 “ {𝐶}) ∧ 𝑥 ∈ (𝐵 “ {𝐶})) ↔ (〈𝐶, 𝑥〉 ∈ 𝐴 ∧ 〈𝐶, 𝑥〉 ∈ 𝐵))) |
11 | 3, 5, 10 | 3bitr4rd 313 | . . 3 ⊢ (𝐶 ∈ 𝑉 → ((𝑥 ∈ (𝐴 “ {𝐶}) ∧ 𝑥 ∈ (𝐵 “ {𝐶})) ↔ 𝑥 ∈ ((𝐴 ∩ 𝐵) “ {𝐶}))) |
12 | 1, 11 | syl5rbb 285 | . 2 ⊢ (𝐶 ∈ 𝑉 → (𝑥 ∈ ((𝐴 ∩ 𝐵) “ {𝐶}) ↔ 𝑥 ∈ ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶})))) |
13 | 12 | eqrdv 2816 | 1 ⊢ (𝐶 ∈ 𝑉 → ((𝐴 ∩ 𝐵) “ {𝐶}) = ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∩ cin 3932 {csn 4557 〈cop 4563 “ cima 5551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 |
This theorem is referenced by: restutopopn 22774 ustuqtop2 22778 |
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