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Theorem xpima 6020
Description: Direct image by a 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 892 . . 3 ((𝐴𝐶) = ∅ ∨ ¬ (𝐴𝐶) = ∅)
2 df-ima 5549 . . . . . . . 8 ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ↾ 𝐶)
3 df-res 5548 . . . . . . . . 9 ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
43rneqi 5788 . . . . . . . 8 ran ((𝐴 × 𝐵) ↾ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
52, 4eqtri 2847 . . . . . . 7 ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
6 inxp 5684 . . . . . . . 8 ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴𝐶) × (𝐵 ∩ V))
76rneqi 5788 . . . . . . 7 ran ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ran ((𝐴𝐶) × (𝐵 ∩ V))
8 inv1 4329 . . . . . . . . 9 (𝐵 ∩ V) = 𝐵
98xpeq2i 5563 . . . . . . . 8 ((𝐴𝐶) × (𝐵 ∩ V)) = ((𝐴𝐶) × 𝐵)
109rneqi 5788 . . . . . . 7 ran ((𝐴𝐶) × (𝐵 ∩ V)) = ran ((𝐴𝐶) × 𝐵)
115, 7, 103eqtri 2851 . . . . . 6 ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴𝐶) × 𝐵)
12 xpeq1 5550 . . . . . . . . 9 ((𝐴𝐶) = ∅ → ((𝐴𝐶) × 𝐵) = (∅ × 𝐵))
13 0xp 5630 . . . . . . . . 9 (∅ × 𝐵) = ∅
1412, 13syl6eq 2875 . . . . . . . 8 ((𝐴𝐶) = ∅ → ((𝐴𝐶) × 𝐵) = ∅)
1514rneqd 5789 . . . . . . 7 ((𝐴𝐶) = ∅ → ran ((𝐴𝐶) × 𝐵) = ran ∅)
16 rn0 5777 . . . . . . 7 ran ∅ = ∅
1715, 16syl6eq 2875 . . . . . 6 ((𝐴𝐶) = ∅ → ran ((𝐴𝐶) × 𝐵) = ∅)
1811, 17syl5eq 2871 . . . . 5 ((𝐴𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = ∅)
1918ancli 552 . . . 4 ((𝐴𝐶) = ∅ → ((𝐴𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = ∅))
20 df-ne 3014 . . . . . . 7 ((𝐴𝐶) ≠ ∅ ↔ ¬ (𝐴𝐶) = ∅)
21 rnxp 6008 . . . . . . 7 ((𝐴𝐶) ≠ ∅ → ran ((𝐴𝐶) × 𝐵) = 𝐵)
2220, 21sylbir 238 . . . . . 6 (¬ (𝐴𝐶) = ∅ → ran ((𝐴𝐶) × 𝐵) = 𝐵)
2311, 22syl5eq 2871 . . . . 5 (¬ (𝐴𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)
2423ancli 552 . . . 4 (¬ (𝐴𝐶) = ∅ → (¬ (𝐴𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = 𝐵))
2519, 24orim12i 906 . . 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 4488 . 2 (((𝐴 × 𝐵) “ 𝐶) = if((𝐴𝐶) = ∅, ∅, 𝐵) ↔ (((𝐴𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = ∅) ∨ (¬ (𝐴𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = 𝐵)))
2826, 27mpbir 234 1 ((𝐴 × 𝐵) “ 𝐶) = if((𝐴𝐶) = ∅, ∅, 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 399  wo 844   = wceq 1538  wne 3013  Vcvv 3479  cin 3917  c0 4274  ifcif 4448   × cxp 5534  ran crn 5537  cres 5538  cima 5539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pr 5311
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-br 5048  df-opab 5110  df-xp 5542  df-rel 5543  df-cnv 5544  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549
This theorem is referenced by:  xpima1  6021  xpima2  6022  imadifxp  30348  bj-xpimasn  34295  bj-imdirco  34509
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