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Theorem imarnf1pr 41624
Description: The image of the range of a function 𝐹 under a function 𝐸 if 𝐹 is a function of 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 6083 . . . . . . . . 9 (𝐸:dom 𝐸𝑅𝐸 Fn dom 𝐸)
21adantl 481 . . . . . . . 8 ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) → 𝐸 Fn dom 𝐸)
32adantr 480 . . . . . . 7 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ (𝑋𝑉𝑌𝑊)) → 𝐸 Fn dom 𝐸)
4 simpll 805 . . . . . . . 8 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ (𝑋𝑉𝑌𝑊)) → 𝐹:{𝑋, 𝑌}⟶dom 𝐸)
5 prid1g 4327 . . . . . . . . . 10 (𝑋𝑉𝑋 ∈ {𝑋, 𝑌})
65adantr 480 . . . . . . . . 9 ((𝑋𝑉𝑌𝑊) → 𝑋 ∈ {𝑋, 𝑌})
76adantl 481 . . . . . . . 8 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ (𝑋𝑉𝑌𝑊)) → 𝑋 ∈ {𝑋, 𝑌})
84, 7ffvelrnd 6400 . . . . . . 7 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ (𝑋𝑉𝑌𝑊)) → (𝐹𝑋) ∈ dom 𝐸)
9 prid2g 4328 . . . . . . . . 9 (𝑌𝑊𝑌 ∈ {𝑋, 𝑌})
109ad2antll 765 . . . . . . . 8 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ (𝑋𝑉𝑌𝑊)) → 𝑌 ∈ {𝑋, 𝑌})
114, 10ffvelrnd 6400 . . . . . . 7 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ (𝑋𝑉𝑌𝑊)) → (𝐹𝑌) ∈ dom 𝐸)
12 fnimapr 6301 . . . . . . 7 ((𝐸 Fn dom 𝐸 ∧ (𝐹𝑋) ∈ dom 𝐸 ∧ (𝐹𝑌) ∈ dom 𝐸) → (𝐸 “ {(𝐹𝑋), (𝐹𝑌)}) = {(𝐸‘(𝐹𝑋)), (𝐸‘(𝐹𝑌))})
133, 8, 11, 12syl3anc 1366 . . . . . 6 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ (𝑋𝑉𝑌𝑊)) → (𝐸 “ {(𝐹𝑋), (𝐹𝑌)}) = {(𝐸‘(𝐹𝑋)), (𝐸‘(𝐹𝑌))})
1413ex 449 . . . . 5 ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) → ((𝑋𝑉𝑌𝑊) → (𝐸 “ {(𝐹𝑋), (𝐹𝑌)}) = {(𝐸‘(𝐹𝑋)), (𝐸‘(𝐹𝑌))}))
1514adantr 480 . . . 4 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵)) → ((𝑋𝑉𝑌𝑊) → (𝐸 “ {(𝐹𝑋), (𝐹𝑌)}) = {(𝐸‘(𝐹𝑋)), (𝐸‘(𝐹𝑌))}))
1615impcom 445 . . 3 (((𝑋𝑉𝑌𝑊) ∧ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵))) → (𝐸 “ {(𝐹𝑋), (𝐹𝑌)}) = {(𝐸‘(𝐹𝑋)), (𝐸‘(𝐹𝑌))})
17 ffn 6083 . . . . . . . . 9 (𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐹 Fn {𝑋, 𝑌})
18 rnfdmpr 41623 . . . . . . . . 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 480 . . . . . . 7 ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) → ((𝑋𝑉𝑌𝑊) → ran 𝐹 = {(𝐹𝑋), (𝐹𝑌)}))
2120adantr 480 . . . . . 6 (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵)) → ((𝑋𝑉𝑌𝑊) → ran 𝐹 = {(𝐹𝑋), (𝐹𝑌)}))
2221impcom 445 . . . . 5 (((𝑋𝑉𝑌𝑊) ∧ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵))) → ran 𝐹 = {(𝐹𝑋), (𝐹𝑌)})
2322eqcomd 2657 . . . 4 (((𝑋𝑉𝑌𝑊) ∧ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵))) → {(𝐹𝑋), (𝐹𝑌)} = ran 𝐹)
2423imaeq2d 5501 . . 3 (((𝑋𝑉𝑌𝑊) ∧ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵))) → (𝐸 “ {(𝐹𝑋), (𝐹𝑌)}) = (𝐸 “ ran 𝐹))
25 preq12 4302 . . . 4 (((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵) → {(𝐸‘(𝐹𝑋)), (𝐸‘(𝐹𝑌))} = {𝐴, 𝐵})
2625ad2antll 765 . . 3 (((𝑋𝑉𝑌𝑊) ∧ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵))) → {(𝐸‘(𝐹𝑋)), (𝐸‘(𝐹𝑌))} = {𝐴, 𝐵})
2716, 24, 263eqtr3d 2693 . 2 (((𝑋𝑉𝑌𝑊) ∧ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵))) → (𝐸 “ ran 𝐹) = {𝐴, 𝐵})
2827ex 449 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 383   = wceq 1523  wcel 2030  {cpr 4212  dom cdm 5143  ran crn 5144  cima 5146   Fn wfn 5921  wf 5922  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934
This theorem is referenced by: (None)
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