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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 3796 | . . 3 ⊢ (𝑥 ∈ ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶})) ↔ (𝑥 ∈ (𝐴 “ {𝐶}) ∧ 𝑥 ∈ (𝐵 “ {𝐶}))) | |
2 | elin 3796 | . . . . 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 | vex 3203 | . . . . 5 ⊢ 𝑥 ∈ V | |
5 | elimasng 5491 | . . . . 5 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ ((𝐴 ∩ 𝐵) “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ (𝐴 ∩ 𝐵))) | |
6 | 4, 5 | mpan2 707 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → (𝑥 ∈ ((𝐴 ∩ 𝐵) “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ (𝐴 ∩ 𝐵))) |
7 | elimasng 5491 | . . . . . 6 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐴 “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ 𝐴)) | |
8 | 4, 7 | mpan2 707 | . . . . 5 ⊢ (𝐶 ∈ 𝑉 → (𝑥 ∈ (𝐴 “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ 𝐴)) |
9 | elimasng 5491 | . . . . . 6 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐵 “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ 𝐵)) | |
10 | 4, 9 | mpan2 707 | . . . . 5 ⊢ (𝐶 ∈ 𝑉 → (𝑥 ∈ (𝐵 “ {𝐶}) ↔ 〈𝐶, 𝑥〉 ∈ 𝐵)) |
11 | 8, 10 | anbi12d 747 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → ((𝑥 ∈ (𝐴 “ {𝐶}) ∧ 𝑥 ∈ (𝐵 “ {𝐶})) ↔ (〈𝐶, 𝑥〉 ∈ 𝐴 ∧ 〈𝐶, 𝑥〉 ∈ 𝐵))) |
12 | 3, 6, 11 | 3bitr4rd 301 | . . 3 ⊢ (𝐶 ∈ 𝑉 → ((𝑥 ∈ (𝐴 “ {𝐶}) ∧ 𝑥 ∈ (𝐵 “ {𝐶})) ↔ 𝑥 ∈ ((𝐴 ∩ 𝐵) “ {𝐶}))) |
13 | 1, 12 | syl5rbb 273 | . 2 ⊢ (𝐶 ∈ 𝑉 → (𝑥 ∈ ((𝐴 ∩ 𝐵) “ {𝐶}) ↔ 𝑥 ∈ ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶})))) |
14 | 13 | eqrdv 2620 | 1 ⊢ (𝐶 ∈ 𝑉 → ((𝐴 ∩ 𝐵) “ {𝐶}) = ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∩ cin 3573 {csn 4177 〈cop 4183 “ cima 5117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 |
This theorem is referenced by: restutopopn 22042 ustuqtop2 22046 |
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