Theorem xpima2 6122

Description: Direct image by a Cartesian product (case of nonempty intersection with the domain). (Contributed by Thierry Arnoux, 16-Dec-2017.)

Assertion

Ref	Expression
xpima2	⊢ ((𝐴 ∩ 𝐶) ≠ ∅ → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)

Proof of Theorem xpima2

Step	Hyp	Ref	Expression
1		xpima 6120	. 2 ⊢ ((𝐴 × 𝐵) “ 𝐶) = if((𝐴 ∩ 𝐶) = ∅, ∅, 𝐵)
2		ifnefalse 4485	. 2 ⊢ ((𝐴 ∩ 𝐶) ≠ ∅ → if((𝐴 ∩ 𝐶) = ∅, ∅, 𝐵) = 𝐵)
3	1, 2	eqtrid 2788	1 ⊢ ((𝐴 ∩ 𝐶) ≠ ∅ → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)

Syntax hints:   → wi 4   = wceq 1540   ≠ wne 2940   ∩ cin 3897  ∅c0 4269  ifcif 4473   × cxp 5618   “ cima 5623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-br 5093  df-opab 5155  df-xp 5626  df-rel 5627  df-cnv 5628  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633

This theorem is referenced by:  xpimasn  6123  ustuqtop1  23499  ustuqtop5  23503  prjcrv0  40732  brtrclfv2  41656  aacllem  46856