Theorem xpimasn 5176

Description: The image of a singleton by a cross product. (Contributed by Thierry Arnoux, 14-Jan-2018.)

Assertion

Ref	Expression
xpimasn	⊢ (𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)

Proof of Theorem xpimasn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		snmg 3784	. . 3 ⊢ (𝑋 ∈ 𝐴 → ∃𝑥 𝑥 ∈ {𝑋})
2		snssi 3811	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → {𝑋} ⊆ 𝐴)
3		dfss1 3408	. . . . . 6 ⊢ ({𝑋} ⊆ 𝐴 ↔ (𝐴 ∩ {𝑋}) = {𝑋})
4	2, 3	sylib 122	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → (𝐴 ∩ {𝑋}) = {𝑋})
5	4	eleq2d 2299	. . . 4 ⊢ (𝑋 ∈ 𝐴 → (𝑥 ∈ (𝐴 ∩ {𝑋}) ↔ 𝑥 ∈ {𝑋}))
6	5	exbidv 1871	. . 3 ⊢ (𝑋 ∈ 𝐴 → (∃𝑥 𝑥 ∈ (𝐴 ∩ {𝑋}) ↔ ∃𝑥 𝑥 ∈ {𝑋}))
7	1, 6	mpbird 167	. 2 ⊢ (𝑋 ∈ 𝐴 → ∃𝑥 𝑥 ∈ (𝐴 ∩ {𝑋}))
8		xpima2m 5175	. 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ {𝑋}) → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)
9	7, 8	syl 14	1 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)

Syntax hints:   → wi 4   = wceq 1395  ∃wex 1538   ∈ wcel 2200   ∩ cin 3196   ⊆ wss 3197  {csn 3666   × cxp 4716   “ cima 4721

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-xp 4724  df-rel 4725  df-cnv 4726  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731

This theorem is referenced by:  imasnopn  14967