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Theorem imarnf1pr 45634
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 6673 . . . . . . . . 9 (𝐸:dom 𝐸𝑅𝐸 Fn dom 𝐸)
21adantl 482 . . . . . . . 8 ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) → 𝐸 Fn dom 𝐸)
32adantr 481 . . . . . . 7 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ (𝑋𝑉𝑌𝑊)) → 𝐸 Fn dom 𝐸)
4 simpll 765 . . . . . . . 8 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ (𝑋𝑉𝑌𝑊)) → 𝐹:{𝑋, 𝑌}⟶dom 𝐸)
5 prid1g 4726 . . . . . . . . . 10 (𝑋𝑉𝑋 ∈ {𝑋, 𝑌})
65adantr 481 . . . . . . . . 9 ((𝑋𝑉𝑌𝑊) → 𝑋 ∈ {𝑋, 𝑌})
76adantl 482 . . . . . . . 8 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ (𝑋𝑉𝑌𝑊)) → 𝑋 ∈ {𝑋, 𝑌})
84, 7ffvelcdmd 7041 . . . . . . 7 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ (𝑋𝑉𝑌𝑊)) → (𝐹𝑋) ∈ dom 𝐸)
9 prid2g 4727 . . . . . . . . 9 (𝑌𝑊𝑌 ∈ {𝑋, 𝑌})
109ad2antll 727 . . . . . . . 8 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ (𝑋𝑉𝑌𝑊)) → 𝑌 ∈ {𝑋, 𝑌})
114, 10ffvelcdmd 7041 . . . . . . 7 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ (𝑋𝑉𝑌𝑊)) → (𝐹𝑌) ∈ dom 𝐸)
12 fnimapr 6930 . . . . . . 7 ((𝐸 Fn dom 𝐸 ∧ (𝐹𝑋) ∈ dom 𝐸 ∧ (𝐹𝑌) ∈ dom 𝐸) → (𝐸 “ {(𝐹𝑋), (𝐹𝑌)}) = {(𝐸‘(𝐹𝑋)), (𝐸‘(𝐹𝑌))})
133, 8, 11, 12syl3anc 1371 . . . . . 6 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ (𝑋𝑉𝑌𝑊)) → (𝐸 “ {(𝐹𝑋), (𝐹𝑌)}) = {(𝐸‘(𝐹𝑋)), (𝐸‘(𝐹𝑌))})
1413ex 413 . . . . 5 ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) → ((𝑋𝑉𝑌𝑊) → (𝐸 “ {(𝐹𝑋), (𝐹𝑌)}) = {(𝐸‘(𝐹𝑋)), (𝐸‘(𝐹𝑌))}))
1514adantr 481 . . . 4 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵)) → ((𝑋𝑉𝑌𝑊) → (𝐸 “ {(𝐹𝑋), (𝐹𝑌)}) = {(𝐸‘(𝐹𝑋)), (𝐸‘(𝐹𝑌))}))
1615impcom 408 . . 3 (((𝑋𝑉𝑌𝑊) ∧ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵))) → (𝐸 “ {(𝐹𝑋), (𝐹𝑌)}) = {(𝐸‘(𝐹𝑋)), (𝐸‘(𝐹𝑌))})
17 ffn 6673 . . . . . . . . 9 (𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐹 Fn {𝑋, 𝑌})
18 rnfdmpr 45633 . . . . . . . . 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 481 . . . . . . 7 ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) → ((𝑋𝑉𝑌𝑊) → ran 𝐹 = {(𝐹𝑋), (𝐹𝑌)}))
2120adantr 481 . . . . . 6 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵)) → ((𝑋𝑉𝑌𝑊) → ran 𝐹 = {(𝐹𝑋), (𝐹𝑌)}))
2221impcom 408 . . . . 5 (((𝑋𝑉𝑌𝑊) ∧ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵))) → ran 𝐹 = {(𝐹𝑋), (𝐹𝑌)})
2322eqcomd 2737 . . . 4 (((𝑋𝑉𝑌𝑊) ∧ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵))) → {(𝐹𝑋), (𝐹𝑌)} = ran 𝐹)
2423imaeq2d 6018 . . 3 (((𝑋𝑉𝑌𝑊) ∧ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵))) → (𝐸 “ {(𝐹𝑋), (𝐹𝑌)}) = (𝐸 “ ran 𝐹))
25 preq12 4701 . . . 4 (((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵) → {(𝐸‘(𝐹𝑋)), (𝐸‘(𝐹𝑌))} = {𝐴, 𝐵})
2625ad2antll 727 . . 3 (((𝑋𝑉𝑌𝑊) ∧ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵))) → {(𝐸‘(𝐹𝑋)), (𝐸‘(𝐹𝑌))} = {𝐴, 𝐵})
2716, 24, 263eqtr3d 2779 . 2 (((𝑋𝑉𝑌𝑊) ∧ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵))) → (𝐸 “ ran 𝐹) = {𝐴, 𝐵})
2827ex 413 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 396   = wceq 1541  wcel 2106  {cpr 4593  dom cdm 5638  ran crn 5639  cima 5641   Fn wfn 6496  wf 6497  cfv 6501
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509
This theorem is referenced by: (None)
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