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Theorem xpima 5480
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 429 . . 3 ((𝐴𝐶) = ∅ ∨ ¬ (𝐴𝐶) = ∅)
2 df-ima 5040 . . . . . . . 8 ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ↾ 𝐶)
3 df-res 5039 . . . . . . . . 9 ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
43rneqi 5259 . . . . . . . 8 ran ((𝐴 × 𝐵) ↾ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
52, 4eqtri 2631 . . . . . . 7 ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
6 inxp 5163 . . . . . . . 8 ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴𝐶) × (𝐵 ∩ V))
76rneqi 5259 . . . . . . 7 ran ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ran ((𝐴𝐶) × (𝐵 ∩ V))
8 inv1 3921 . . . . . . . . 9 (𝐵 ∩ V) = 𝐵
98xpeq2i 5049 . . . . . . . 8 ((𝐴𝐶) × (𝐵 ∩ V)) = ((𝐴𝐶) × 𝐵)
109rneqi 5259 . . . . . . 7 ran ((𝐴𝐶) × (𝐵 ∩ V)) = ran ((𝐴𝐶) × 𝐵)
115, 7, 103eqtri 2635 . . . . . 6 ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴𝐶) × 𝐵)
12 xpeq1 5041 . . . . . . . . 9 ((𝐴𝐶) = ∅ → ((𝐴𝐶) × 𝐵) = (∅ × 𝐵))
13 0xp 5111 . . . . . . . . 9 (∅ × 𝐵) = ∅
1412, 13syl6eq 2659 . . . . . . . 8 ((𝐴𝐶) = ∅ → ((𝐴𝐶) × 𝐵) = ∅)
1514rneqd 5260 . . . . . . 7 ((𝐴𝐶) = ∅ → ran ((𝐴𝐶) × 𝐵) = ran ∅)
16 rn0 5284 . . . . . . 7 ran ∅ = ∅
1715, 16syl6eq 2659 . . . . . 6 ((𝐴𝐶) = ∅ → ran ((𝐴𝐶) × 𝐵) = ∅)
1811, 17syl5eq 2655 . . . . 5 ((𝐴𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = ∅)
1918ancli 571 . . . 4 ((𝐴𝐶) = ∅ → ((𝐴𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = ∅))
20 df-ne 2781 . . . . . . 7 ((𝐴𝐶) ≠ ∅ ↔ ¬ (𝐴𝐶) = ∅)
21 rnxp 5468 . . . . . . 7 ((𝐴𝐶) ≠ ∅ → ran ((𝐴𝐶) × 𝐵) = 𝐵)
2220, 21sylbir 223 . . . . . 6 (¬ (𝐴𝐶) = ∅ → ran ((𝐴𝐶) × 𝐵) = 𝐵)
2311, 22syl5eq 2655 . . . . 5 (¬ (𝐴𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)
2423ancli 571 . . . 4 (¬ (𝐴𝐶) = ∅ → (¬ (𝐴𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = 𝐵))
2519, 24orim12i 536 . . 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 4075 . 2 (((𝐴 × 𝐵) “ 𝐶) = if((𝐴𝐶) = ∅, ∅, 𝐵) ↔ (((𝐴𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = ∅) ∨ (¬ (𝐴𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = 𝐵)))
2826, 27mpbir 219 1 ((𝐴 × 𝐵) “ 𝐶) = if((𝐴𝐶) = ∅, ∅, 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 381  wa 382   = wceq 1474  wne 2779  Vcvv 3172  cin 3538  c0 3873  ifcif 4035   × cxp 5025  ran crn 5028  cres 5029  cima 5030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-xp 5033  df-rel 5034  df-cnv 5035  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040
This theorem is referenced by:  xpima1  5481  xpima2  5482  imadifxp  28589  bj-xpimasn  31918
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