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Theorem rnfdmpr 42030
Description: The range of a one-to-one function 𝐹 of an unordered pair into a set is the unordered pair of the function values. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
Assertion
Ref Expression
rnfdmpr ((𝑋𝑉𝑌𝑊) → (𝐹 Fn {𝑋, 𝑌} → ran 𝐹 = {(𝐹𝑋), (𝐹𝑌)}))

Proof of Theorem rnfdmpr
Dummy variables 𝑥 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnrnfv 6431 . . . 4 (𝐹 Fn {𝑋, 𝑌} → ran 𝐹 = {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹𝑖)})
21adantl 473 . . 3 (((𝑋𝑉𝑌𝑊) ∧ 𝐹 Fn {𝑋, 𝑌}) → ran 𝐹 = {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹𝑖)})
3 fveq2 6375 . . . . . . . 8 (𝑖 = 𝑋 → (𝐹𝑖) = (𝐹𝑋))
43eqeq2d 2775 . . . . . . 7 (𝑖 = 𝑋 → (𝑥 = (𝐹𝑖) ↔ 𝑥 = (𝐹𝑋)))
54abbidv 2884 . . . . . 6 (𝑖 = 𝑋 → {𝑥𝑥 = (𝐹𝑖)} = {𝑥𝑥 = (𝐹𝑋)})
6 fveq2 6375 . . . . . . . 8 (𝑖 = 𝑌 → (𝐹𝑖) = (𝐹𝑌))
76eqeq2d 2775 . . . . . . 7 (𝑖 = 𝑌 → (𝑥 = (𝐹𝑖) ↔ 𝑥 = (𝐹𝑌)))
87abbidv 2884 . . . . . 6 (𝑖 = 𝑌 → {𝑥𝑥 = (𝐹𝑖)} = {𝑥𝑥 = (𝐹𝑌)})
95, 8iunxprg 4764 . . . . 5 ((𝑋𝑉𝑌𝑊) → 𝑖 ∈ {𝑋, 𝑌} {𝑥𝑥 = (𝐹𝑖)} = ({𝑥𝑥 = (𝐹𝑋)} ∪ {𝑥𝑥 = (𝐹𝑌)}))
109adantr 472 . . . 4 (((𝑋𝑉𝑌𝑊) ∧ 𝐹 Fn {𝑋, 𝑌}) → 𝑖 ∈ {𝑋, 𝑌} {𝑥𝑥 = (𝐹𝑖)} = ({𝑥𝑥 = (𝐹𝑋)} ∪ {𝑥𝑥 = (𝐹𝑌)}))
11 iunab 4722 . . . 4 𝑖 ∈ {𝑋, 𝑌} {𝑥𝑥 = (𝐹𝑖)} = {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹𝑖)}
12 df-sn 4335 . . . . . . 7 {(𝐹𝑋)} = {𝑥𝑥 = (𝐹𝑋)}
1312eqcomi 2774 . . . . . 6 {𝑥𝑥 = (𝐹𝑋)} = {(𝐹𝑋)}
14 df-sn 4335 . . . . . . 7 {(𝐹𝑌)} = {𝑥𝑥 = (𝐹𝑌)}
1514eqcomi 2774 . . . . . 6 {𝑥𝑥 = (𝐹𝑌)} = {(𝐹𝑌)}
1613, 15uneq12i 3927 . . . . 5 ({𝑥𝑥 = (𝐹𝑋)} ∪ {𝑥𝑥 = (𝐹𝑌)}) = ({(𝐹𝑋)} ∪ {(𝐹𝑌)})
17 df-pr 4337 . . . . 5 {(𝐹𝑋), (𝐹𝑌)} = ({(𝐹𝑋)} ∪ {(𝐹𝑌)})
1816, 17eqtr4i 2790 . . . 4 ({𝑥𝑥 = (𝐹𝑋)} ∪ {𝑥𝑥 = (𝐹𝑌)}) = {(𝐹𝑋), (𝐹𝑌)}
1910, 11, 183eqtr3g 2822 . . 3 (((𝑋𝑉𝑌𝑊) ∧ 𝐹 Fn {𝑋, 𝑌}) → {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹𝑖)} = {(𝐹𝑋), (𝐹𝑌)})
202, 19eqtrd 2799 . 2 (((𝑋𝑉𝑌𝑊) ∧ 𝐹 Fn {𝑋, 𝑌}) → ran 𝐹 = {(𝐹𝑋), (𝐹𝑌)})
2120ex 401 1 ((𝑋𝑉𝑌𝑊) → (𝐹 Fn {𝑋, 𝑌} → ran 𝐹 = {(𝐹𝑋), (𝐹𝑌)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  {cab 2751  wrex 3056  cun 3730  {csn 4334  {cpr 4336   ciun 4676  ran crn 5278   Fn wfn 6063  cfv 6068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-iota 6031  df-fun 6070  df-fn 6071  df-fv 6076
This theorem is referenced by:  imarnf1pr  42031
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