Theorem xpima2m 4711

Description: The image by a cross product. (Contributed by Thierry Arnoux, 16-Dec-2017.)

Assertion

Ref	Expression
xpima2m	⊢ (∃x x ∈ (A ∩ 𝐶) → ((A × B) “ 𝐶) = B)

Distinct variable groups:   x,A   x,𝐶

Allowed substitution hint:   B(x)

Proof of Theorem xpima2m

Step	Hyp	Ref	Expression
1		df-ima 4301	. . . 4 ⊢ ((A × B) “ 𝐶) = ran ((A × B) ↾ 𝐶)
2		df-res 4300	. . . . 5 ⊢ ((A × B) ↾ 𝐶) = ((A × B) ∩ (𝐶 × V))
3	2	rneqi 4505	. . . 4 ⊢ ran ((A × B) ↾ 𝐶) = ran ((A × B) ∩ (𝐶 × V))
4		inxp 4413	. . . . 5 ⊢ ((A × B) ∩ (𝐶 × V)) = ((A ∩ 𝐶) × (B ∩ V))
5	4	rneqi 4505	. . . 4 ⊢ ran ((A × B) ∩ (𝐶 × V)) = ran ((A ∩ 𝐶) × (B ∩ V))
6	1, 3, 5	3eqtri 2061	. . 3 ⊢ ((A × B) “ 𝐶) = ran ((A ∩ 𝐶) × (B ∩ V))
7		rnxpm 4695	. . 3 ⊢ (∃x x ∈ (A ∩ 𝐶) → ran ((A ∩ 𝐶) × (B ∩ V)) = (B ∩ V))
8	6, 7	syl5eq 2081	. 2 ⊢ (∃x x ∈ (A ∩ 𝐶) → ((A × B) “ 𝐶) = (B ∩ V))
9		inv1 3247	. 2 ⊢ (B ∩ V) = B
10	8, 9	syl6eq 2085	1 ⊢ (∃x x ∈ (A ∩ 𝐶) → ((A × B) “ 𝐶) = B)

Syntax hints:   → wi 4   = wceq 1242  ∃wex 1378   ∈ wcel 1390  Vcvv 2551   ∩ cin 2910   × cxp 4286  ran crn 4289   ↾ cres 4290   “ 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:  xpimasn  4712