Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpima Structured version   Visualization version   GIF version

Theorem xpima 5495
 Description: The image by a constant function (or other Cartesian product). (Contributed by Thierry Arnoux, 4-Feb-2017.)
Assertion
Ref Expression
xpima ((𝐴 × 𝐵) “ 𝐶) = if((𝐴𝐶) = ∅, ∅, 𝐵)

Proof of Theorem xpima
StepHypRef Expression
1 exmid 430 . . 3 ((𝐴𝐶) = ∅ ∨ ¬ (𝐴𝐶) = ∅)
2 df-ima 5051 . . . . . . . 8 ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ↾ 𝐶)
3 df-res 5050 . . . . . . . . 9 ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
43rneqi 5273 . . . . . . . 8 ran ((𝐴 × 𝐵) ↾ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
52, 4eqtri 2632 . . . . . . 7 ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
6 inxp 5176 . . . . . . . 8 ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴𝐶) × (𝐵 ∩ V))
76rneqi 5273 . . . . . . 7 ran ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ran ((𝐴𝐶) × (𝐵 ∩ V))
8 inv1 3922 . . . . . . . . 9 (𝐵 ∩ V) = 𝐵
98xpeq2i 5060 . . . . . . . 8 ((𝐴𝐶) × (𝐵 ∩ V)) = ((𝐴𝐶) × 𝐵)
109rneqi 5273 . . . . . . 7 ran ((𝐴𝐶) × (𝐵 ∩ V)) = ran ((𝐴𝐶) × 𝐵)
115, 7, 103eqtri 2636 . . . . . 6 ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴𝐶) × 𝐵)
12 xpeq1 5052 . . . . . . . . 9 ((𝐴𝐶) = ∅ → ((𝐴𝐶) × 𝐵) = (∅ × 𝐵))
13 0xp 5122 . . . . . . . . 9 (∅ × 𝐵) = ∅
1412, 13syl6eq 2660 . . . . . . . 8 ((𝐴𝐶) = ∅ → ((𝐴𝐶) × 𝐵) = ∅)
1514rneqd 5274 . . . . . . 7 ((𝐴𝐶) = ∅ → ran ((𝐴𝐶) × 𝐵) = ran ∅)
16 rn0 5298 . . . . . . 7 ran ∅ = ∅
1715, 16syl6eq 2660 . . . . . 6 ((𝐴𝐶) = ∅ → ran ((𝐴𝐶) × 𝐵) = ∅)
1811, 17syl5eq 2656 . . . . 5 ((𝐴𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = ∅)
1918ancli 572 . . . 4 ((𝐴𝐶) = ∅ → ((𝐴𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = ∅))
20 df-ne 2782 . . . . . . 7 ((𝐴𝐶) ≠ ∅ ↔ ¬ (𝐴𝐶) = ∅)
21 rnxp 5483 . . . . . . 7 ((𝐴𝐶) ≠ ∅ → ran ((𝐴𝐶) × 𝐵) = 𝐵)
2220, 21sylbir 224 . . . . . 6 (¬ (𝐴𝐶) = ∅ → ran ((𝐴𝐶) × 𝐵) = 𝐵)
2311, 22syl5eq 2656 . . . . 5 (¬ (𝐴𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)
2423ancli 572 . . . 4 (¬ (𝐴𝐶) = ∅ → (¬ (𝐴𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = 𝐵))
2519, 24orim12i 537 . . 3 (((𝐴𝐶) = ∅ ∨ ¬ (𝐴𝐶) = ∅) → (((𝐴𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = ∅) ∨ (¬ (𝐴𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = 𝐵)))
261, 25ax-mp 5 . 2 (((𝐴𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = ∅) ∨ (¬ (𝐴𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = 𝐵))
27 eqif 4076 . 2 (((𝐴 × 𝐵) “ 𝐶) = if((𝐴𝐶) = ∅, ∅, 𝐵) ↔ (((𝐴𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = ∅) ∨ (¬ (𝐴𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = 𝐵)))
2826, 27mpbir 220 1 ((𝐴 × 𝐵) “ 𝐶) = if((𝐴𝐶) = ∅, ∅, 𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 382   ∧ wa 383   = wceq 1475   ≠ wne 2780  Vcvv 3173   ∩ cin 3539  ∅c0 3874  ifcif 4036   × cxp 5036  ran crn 5039   ↾ cres 5040   “ cima 5041 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051 This theorem is referenced by:  xpima1  5496  xpima2  5497  imadifxp  28796  bj-xpimasn  32135
 Copyright terms: Public domain W3C validator