Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  imarnf1pr Structured version   Visualization version   GIF version

Theorem imarnf1pr 44206
 Description: The image of the range of a function 𝐹 under a function 𝐸 if 𝐹 is a function from a pair into the domain of 𝐸. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
Assertion
Ref Expression
imarnf1pr ((𝑋𝑉𝑌𝑊) → (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵)) → (𝐸 “ ran 𝐹) = {𝐴, 𝐵}))

Proof of Theorem imarnf1pr
StepHypRef Expression
1 ffn 6498 . . . . . . . . 9 (𝐸:dom 𝐸𝑅𝐸 Fn dom 𝐸)
21adantl 485 . . . . . . . 8 ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) → 𝐸 Fn dom 𝐸)
32adantr 484 . . . . . . 7 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ (𝑋𝑉𝑌𝑊)) → 𝐸 Fn dom 𝐸)
4 simpll 766 . . . . . . . 8 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ (𝑋𝑉𝑌𝑊)) → 𝐹:{𝑋, 𝑌}⟶dom 𝐸)
5 prid1g 4653 . . . . . . . . . 10 (𝑋𝑉𝑋 ∈ {𝑋, 𝑌})
65adantr 484 . . . . . . . . 9 ((𝑋𝑉𝑌𝑊) → 𝑋 ∈ {𝑋, 𝑌})
76adantl 485 . . . . . . . 8 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ (𝑋𝑉𝑌𝑊)) → 𝑋 ∈ {𝑋, 𝑌})
84, 7ffvelrnd 6843 . . . . . . 7 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ (𝑋𝑉𝑌𝑊)) → (𝐹𝑋) ∈ dom 𝐸)
9 prid2g 4654 . . . . . . . . 9 (𝑌𝑊𝑌 ∈ {𝑋, 𝑌})
109ad2antll 728 . . . . . . . 8 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ (𝑋𝑉𝑌𝑊)) → 𝑌 ∈ {𝑋, 𝑌})
114, 10ffvelrnd 6843 . . . . . . 7 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ (𝑋𝑉𝑌𝑊)) → (𝐹𝑌) ∈ dom 𝐸)
12 fnimapr 6736 . . . . . . 7 ((𝐸 Fn dom 𝐸 ∧ (𝐹𝑋) ∈ dom 𝐸 ∧ (𝐹𝑌) ∈ dom 𝐸) → (𝐸 “ {(𝐹𝑋), (𝐹𝑌)}) = {(𝐸‘(𝐹𝑋)), (𝐸‘(𝐹𝑌))})
133, 8, 11, 12syl3anc 1368 . . . . . 6 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ (𝑋𝑉𝑌𝑊)) → (𝐸 “ {(𝐹𝑋), (𝐹𝑌)}) = {(𝐸‘(𝐹𝑋)), (𝐸‘(𝐹𝑌))})
1413ex 416 . . . . 5 ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) → ((𝑋𝑉𝑌𝑊) → (𝐸 “ {(𝐹𝑋), (𝐹𝑌)}) = {(𝐸‘(𝐹𝑋)), (𝐸‘(𝐹𝑌))}))
1514adantr 484 . . . 4 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵)) → ((𝑋𝑉𝑌𝑊) → (𝐸 “ {(𝐹𝑋), (𝐹𝑌)}) = {(𝐸‘(𝐹𝑋)), (𝐸‘(𝐹𝑌))}))
1615impcom 411 . . 3 (((𝑋𝑉𝑌𝑊) ∧ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵))) → (𝐸 “ {(𝐹𝑋), (𝐹𝑌)}) = {(𝐸‘(𝐹𝑋)), (𝐸‘(𝐹𝑌))})
17 ffn 6498 . . . . . . . . 9 (𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐹 Fn {𝑋, 𝑌})
18 rnfdmpr 44205 . . . . . . . . 9 ((𝑋𝑉𝑌𝑊) → (𝐹 Fn {𝑋, 𝑌} → ran 𝐹 = {(𝐹𝑋), (𝐹𝑌)}))
1917, 18syl5com 31 . . . . . . . 8 (𝐹:{𝑋, 𝑌}⟶dom 𝐸 → ((𝑋𝑉𝑌𝑊) → ran 𝐹 = {(𝐹𝑋), (𝐹𝑌)}))
2019adantr 484 . . . . . . 7 ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) → ((𝑋𝑉𝑌𝑊) → ran 𝐹 = {(𝐹𝑋), (𝐹𝑌)}))
2120adantr 484 . . . . . 6 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵)) → ((𝑋𝑉𝑌𝑊) → ran 𝐹 = {(𝐹𝑋), (𝐹𝑌)}))
2221impcom 411 . . . . 5 (((𝑋𝑉𝑌𝑊) ∧ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵))) → ran 𝐹 = {(𝐹𝑋), (𝐹𝑌)})
2322eqcomd 2764 . . . 4 (((𝑋𝑉𝑌𝑊) ∧ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵))) → {(𝐹𝑋), (𝐹𝑌)} = ran 𝐹)
2423imaeq2d 5901 . . 3 (((𝑋𝑉𝑌𝑊) ∧ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵))) → (𝐸 “ {(𝐹𝑋), (𝐹𝑌)}) = (𝐸 “ ran 𝐹))
25 preq12 4628 . . . 4 (((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵) → {(𝐸‘(𝐹𝑋)), (𝐸‘(𝐹𝑌))} = {𝐴, 𝐵})
2625ad2antll 728 . . 3 (((𝑋𝑉𝑌𝑊) ∧ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵))) → {(𝐸‘(𝐹𝑋)), (𝐸‘(𝐹𝑌))} = {𝐴, 𝐵})
2716, 24, 263eqtr3d 2801 . 2 (((𝑋𝑉𝑌𝑊) ∧ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵))) → (𝐸 “ ran 𝐹) = {𝐴, 𝐵})
2827ex 416 1 ((𝑋𝑉𝑌𝑊) → (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵)) → (𝐸 “ ran 𝐹) = {𝐴, 𝐵}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cpr 4524  dom cdm 5524  ran crn 5525   “ cima 5527   Fn wfn 6330  ⟶wf 6331  ‘cfv 6335 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-fv 6343 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator