Theorem xpimasn 4712

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 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)

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-bnd 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)