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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 3477 | . . 3 ⊢ (𝑋 ∈ A → ∃x x ∈ {𝑋}) | |
2 | snssi 3499 | . . . . . 6 ⊢ (𝑋 ∈ A → {𝑋} ⊆ A) | |
3 | dfss1 3135 | . . . . . 6 ⊢ ({𝑋} ⊆ A ↔ (A ∩ {𝑋}) = {𝑋}) | |
4 | 2, 3 | sylib 127 | . . . . 5 ⊢ (𝑋 ∈ A → (A ∩ {𝑋}) = {𝑋}) |
5 | 4 | eleq2d 2104 | . . . 4 ⊢ (𝑋 ∈ A → (x ∈ (A ∩ {𝑋}) ↔ x ∈ {𝑋})) |
6 | 5 | exbidv 1703 | . . 3 ⊢ (𝑋 ∈ A → (∃x x ∈ (A ∩ {𝑋}) ↔ ∃x x ∈ {𝑋})) |
7 | 1, 6 | mpbird 156 | . 2 ⊢ (𝑋 ∈ A → ∃x x ∈ (A ∩ {𝑋})) |
8 | xpima2m 4711 | . 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 1242 ∃wex 1378 ∈ wcel 1390 ∩ cin 2910 ⊆ wss 2911 {csn 3367 × cxp 4286 “ cima 4291 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-xp 4294 df-rel 4295 df-cnv 4296 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 |
This theorem is referenced by: (None) |
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