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Theorem imarnf1pr 45990
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 6718 . . . . . . . . 9 (𝐸:dom 𝐸𝑅𝐸 Fn dom 𝐸)
21adantl 483 . . . . . . . 8 ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) → 𝐸 Fn dom 𝐸)
32adantr 482 . . . . . . 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 4765 . . . . . . . . . 10 (𝑋𝑉𝑋 ∈ {𝑋, 𝑌})
65adantr 482 . . . . . . . . 9 ((𝑋𝑉𝑌𝑊) → 𝑋 ∈ {𝑋, 𝑌})
76adantl 483 . . . . . . . 8 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ (𝑋𝑉𝑌𝑊)) → 𝑋 ∈ {𝑋, 𝑌})
84, 7ffvelcdmd 7088 . . . . . . 7 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ (𝑋𝑉𝑌𝑊)) → (𝐹𝑋) ∈ dom 𝐸)
9 prid2g 4766 . . . . . . . . 9 (𝑌𝑊𝑌 ∈ {𝑋, 𝑌})
109ad2antll 728 . . . . . . . 8 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ (𝑋𝑉𝑌𝑊)) → 𝑌 ∈ {𝑋, 𝑌})
114, 10ffvelcdmd 7088 . . . . . . 7 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ (𝑋𝑉𝑌𝑊)) → (𝐹𝑌) ∈ dom 𝐸)
12 fnimapr 6976 . . . . . . 7 ((𝐸 Fn dom 𝐸 ∧ (𝐹𝑋) ∈ dom 𝐸 ∧ (𝐹𝑌) ∈ dom 𝐸) → (𝐸 “ {(𝐹𝑋), (𝐹𝑌)}) = {(𝐸‘(𝐹𝑋)), (𝐸‘(𝐹𝑌))})
133, 8, 11, 12syl3anc 1372 . . . . . 6 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ (𝑋𝑉𝑌𝑊)) → (𝐸 “ {(𝐹𝑋), (𝐹𝑌)}) = {(𝐸‘(𝐹𝑋)), (𝐸‘(𝐹𝑌))})
1413ex 414 . . . . 5 ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) → ((𝑋𝑉𝑌𝑊) → (𝐸 “ {(𝐹𝑋), (𝐹𝑌)}) = {(𝐸‘(𝐹𝑋)), (𝐸‘(𝐹𝑌))}))
1514adantr 482 . . . 4 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵)) → ((𝑋𝑉𝑌𝑊) → (𝐸 “ {(𝐹𝑋), (𝐹𝑌)}) = {(𝐸‘(𝐹𝑋)), (𝐸‘(𝐹𝑌))}))
1615impcom 409 . . 3 (((𝑋𝑉𝑌𝑊) ∧ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵))) → (𝐸 “ {(𝐹𝑋), (𝐹𝑌)}) = {(𝐸‘(𝐹𝑋)), (𝐸‘(𝐹𝑌))})
17 ffn 6718 . . . . . . . . 9 (𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐹 Fn {𝑋, 𝑌})
18 rnfdmpr 45989 . . . . . . . . 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 482 . . . . . . 7 ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) → ((𝑋𝑉𝑌𝑊) → ran 𝐹 = {(𝐹𝑋), (𝐹𝑌)}))
2120adantr 482 . . . . . 6 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵)) → ((𝑋𝑉𝑌𝑊) → ran 𝐹 = {(𝐹𝑋), (𝐹𝑌)}))
2221impcom 409 . . . . 5 (((𝑋𝑉𝑌𝑊) ∧ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵))) → ran 𝐹 = {(𝐹𝑋), (𝐹𝑌)})
2322eqcomd 2739 . . . 4 (((𝑋𝑉𝑌𝑊) ∧ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵))) → {(𝐹𝑋), (𝐹𝑌)} = ran 𝐹)
2423imaeq2d 6060 . . 3 (((𝑋𝑉𝑌𝑊) ∧ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵))) → (𝐸 “ {(𝐹𝑋), (𝐹𝑌)}) = (𝐸 “ ran 𝐹))
25 preq12 4740 . . . 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 2781 . 2 (((𝑋𝑉𝑌𝑊) ∧ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵))) → (𝐸 “ ran 𝐹) = {𝐴, 𝐵})
2827ex 414 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 397   = wceq 1542  wcel 2107  {cpr 4631  dom cdm 5677  ran crn 5678  cima 5680   Fn wfn 6539  wf 6540  cfv 6544
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 5300  ax-nul 5307  ax-pr 5428
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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552
This theorem is referenced by: (None)
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