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Mirrors > Home > ILE Home > Th. List > xpimasn | GIF version |
Description: The image of a singleton by a cross product. (Contributed by Thierry Arnoux, 14-Jan-2018.) |
Ref | Expression |
---|---|
xpimasn | ⊢ (𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snmg 3722 | . . 3 ⊢ (𝑋 ∈ 𝐴 → ∃𝑥 𝑥 ∈ {𝑋}) | |
2 | snssi 3748 | . . . . . 6 ⊢ (𝑋 ∈ 𝐴 → {𝑋} ⊆ 𝐴) | |
3 | dfss1 3351 | . . . . . 6 ⊢ ({𝑋} ⊆ 𝐴 ↔ (𝐴 ∩ {𝑋}) = {𝑋}) | |
4 | 2, 3 | sylib 122 | . . . . 5 ⊢ (𝑋 ∈ 𝐴 → (𝐴 ∩ {𝑋}) = {𝑋}) |
5 | 4 | eleq2d 2257 | . . . 4 ⊢ (𝑋 ∈ 𝐴 → (𝑥 ∈ (𝐴 ∩ {𝑋}) ↔ 𝑥 ∈ {𝑋})) |
6 | 5 | exbidv 1835 | . . 3 ⊢ (𝑋 ∈ 𝐴 → (∃𝑥 𝑥 ∈ (𝐴 ∩ {𝑋}) ↔ ∃𝑥 𝑥 ∈ {𝑋})) |
7 | 1, 6 | mpbird 167 | . 2 ⊢ (𝑋 ∈ 𝐴 → ∃𝑥 𝑥 ∈ (𝐴 ∩ {𝑋})) |
8 | xpima2m 5088 | . 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ {𝑋}) → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵) | |
9 | 7, 8 | syl 14 | 1 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1363 ∃wex 1502 ∈ wcel 2158 ∩ cin 3140 ⊆ wss 3141 {csn 3604 × cxp 4636 “ cima 4641 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-br 4016 df-opab 4077 df-xp 4644 df-rel 4645 df-cnv 4646 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 |
This theorem is referenced by: imasnopn 14070 |
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