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Theorem xpima 6138
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 894 . . 3 ((𝐴𝐶) = ∅ ∨ ¬ (𝐴𝐶) = ∅)
2 df-ima 5650 . . . . . . . 8 ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ↾ 𝐶)
3 df-res 5649 . . . . . . . . 9 ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
43rneqi 5896 . . . . . . . 8 ran ((𝐴 × 𝐵) ↾ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
52, 4eqtri 2761 . . . . . . 7 ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
6 inxp 5792 . . . . . . . 8 ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴𝐶) × (𝐵 ∩ V))
76rneqi 5896 . . . . . . 7 ran ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ran ((𝐴𝐶) × (𝐵 ∩ V))
8 inv1 4358 . . . . . . . . 9 (𝐵 ∩ V) = 𝐵
98xpeq2i 5664 . . . . . . . 8 ((𝐴𝐶) × (𝐵 ∩ V)) = ((𝐴𝐶) × 𝐵)
109rneqi 5896 . . . . . . 7 ran ((𝐴𝐶) × (𝐵 ∩ V)) = ran ((𝐴𝐶) × 𝐵)
115, 7, 103eqtri 2765 . . . . . 6 ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴𝐶) × 𝐵)
12 xpeq1 5651 . . . . . . . . 9 ((𝐴𝐶) = ∅ → ((𝐴𝐶) × 𝐵) = (∅ × 𝐵))
13 0xp 5734 . . . . . . . . 9 (∅ × 𝐵) = ∅
1412, 13eqtrdi 2789 . . . . . . . 8 ((𝐴𝐶) = ∅ → ((𝐴𝐶) × 𝐵) = ∅)
1514rneqd 5897 . . . . . . 7 ((𝐴𝐶) = ∅ → ran ((𝐴𝐶) × 𝐵) = ran ∅)
16 rn0 5885 . . . . . . 7 ran ∅ = ∅
1715, 16eqtrdi 2789 . . . . . 6 ((𝐴𝐶) = ∅ → ran ((𝐴𝐶) × 𝐵) = ∅)
1811, 17eqtrid 2785 . . . . 5 ((𝐴𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = ∅)
1918ancli 550 . . . 4 ((𝐴𝐶) = ∅ → ((𝐴𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = ∅))
20 df-ne 2941 . . . . . . 7 ((𝐴𝐶) ≠ ∅ ↔ ¬ (𝐴𝐶) = ∅)
21 rnxp 6126 . . . . . . 7 ((𝐴𝐶) ≠ ∅ → ran ((𝐴𝐶) × 𝐵) = 𝐵)
2220, 21sylbir 234 . . . . . 6 (¬ (𝐴𝐶) = ∅ → ran ((𝐴𝐶) × 𝐵) = 𝐵)
2311, 22eqtrid 2785 . . . . 5 (¬ (𝐴𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)
2423ancli 550 . . . 4 (¬ (𝐴𝐶) = ∅ → (¬ (𝐴𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = 𝐵))
2519, 24orim12i 908 . . 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 4531 . 2 (((𝐴 × 𝐵) “ 𝐶) = if((𝐴𝐶) = ∅, ∅, 𝐵) ↔ (((𝐴𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = ∅) ∨ (¬ (𝐴𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = 𝐵)))
2826, 27mpbir 230 1 ((𝐴 × 𝐵) “ 𝐶) = if((𝐴𝐶) = ∅, ∅, 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 397  wo 846   = wceq 1542  wne 2940  Vcvv 3447  cin 3913  c0 4286  ifcif 4490   × cxp 5635  ran crn 5638  cres 5639  cima 5640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-rel 5644  df-cnv 5645  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650
This theorem is referenced by:  xpima1  6139  xpima2  6140  imadifxp  31572  bj-xpimasn  35476  bj-imdirco  35711
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