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Theorem xpima 5734
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 5279 . . . . . . . 8 ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ↾ 𝐶)
3 df-res 5278 . . . . . . . . 9 ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
43rneqi 5507 . . . . . . . 8 ran ((𝐴 × 𝐵) ↾ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
52, 4eqtri 2782 . . . . . . 7 ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
6 inxp 5410 . . . . . . . 8 ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴𝐶) × (𝐵 ∩ V))
76rneqi 5507 . . . . . . 7 ran ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ran ((𝐴𝐶) × (𝐵 ∩ V))
8 inv1 4113 . . . . . . . . 9 (𝐵 ∩ V) = 𝐵
98xpeq2i 5293 . . . . . . . 8 ((𝐴𝐶) × (𝐵 ∩ V)) = ((𝐴𝐶) × 𝐵)
109rneqi 5507 . . . . . . 7 ran ((𝐴𝐶) × (𝐵 ∩ V)) = ran ((𝐴𝐶) × 𝐵)
115, 7, 103eqtri 2786 . . . . . 6 ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴𝐶) × 𝐵)
12 xpeq1 5280 . . . . . . . . 9 ((𝐴𝐶) = ∅ → ((𝐴𝐶) × 𝐵) = (∅ × 𝐵))
13 0xp 5356 . . . . . . . . 9 (∅ × 𝐵) = ∅
1412, 13syl6eq 2810 . . . . . . . 8 ((𝐴𝐶) = ∅ → ((𝐴𝐶) × 𝐵) = ∅)
1514rneqd 5508 . . . . . . 7 ((𝐴𝐶) = ∅ → ran ((𝐴𝐶) × 𝐵) = ran ∅)
16 rn0 5532 . . . . . . 7 ran ∅ = ∅
1715, 16syl6eq 2810 . . . . . 6 ((𝐴𝐶) = ∅ → ran ((𝐴𝐶) × 𝐵) = ∅)
1811, 17syl5eq 2806 . . . . 5 ((𝐴𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = ∅)
1918ancli 575 . . . 4 ((𝐴𝐶) = ∅ → ((𝐴𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = ∅))
20 df-ne 2933 . . . . . . 7 ((𝐴𝐶) ≠ ∅ ↔ ¬ (𝐴𝐶) = ∅)
21 rnxp 5722 . . . . . . 7 ((𝐴𝐶) ≠ ∅ → ran ((𝐴𝐶) × 𝐵) = 𝐵)
2220, 21sylbir 225 . . . . . 6 (¬ (𝐴𝐶) = ∅ → ran ((𝐴𝐶) × 𝐵) = 𝐵)
2311, 22syl5eq 2806 . . . . 5 (¬ (𝐴𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)
2423ancli 575 . . . 4 (¬ (𝐴𝐶) = ∅ → (¬ (𝐴𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = 𝐵))
2519, 24orim12i 539 . . 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 4270 . 2 (((𝐴 × 𝐵) “ 𝐶) = if((𝐴𝐶) = ∅, ∅, 𝐵) ↔ (((𝐴𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = ∅) ∨ (¬ (𝐴𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = 𝐵)))
2826, 27mpbir 221 1 ((𝐴 × 𝐵) “ 𝐶) = if((𝐴𝐶) = ∅, ∅, 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 382  wa 383   = wceq 1632  wne 2932  Vcvv 3340  cin 3714  c0 4058  ifcif 4230   × cxp 5264  ran crn 5267  cres 5268  cima 5269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279
This theorem is referenced by:  xpima1  5735  xpima2  5736  imadifxp  29721  bj-xpimasn  33248
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