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Theorem rnfdmpr 43357
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 6718 . . . 4 (𝐹 Fn {𝑋, 𝑌} → ran 𝐹 = {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹𝑖)})
21adantl 482 . . 3 (((𝑋𝑉𝑌𝑊) ∧ 𝐹 Fn {𝑋, 𝑌}) → ran 𝐹 = {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹𝑖)})
3 fveq2 6663 . . . . . . . 8 (𝑖 = 𝑋 → (𝐹𝑖) = (𝐹𝑋))
43eqeq2d 2829 . . . . . . 7 (𝑖 = 𝑋 → (𝑥 = (𝐹𝑖) ↔ 𝑥 = (𝐹𝑋)))
54abbidv 2882 . . . . . 6 (𝑖 = 𝑋 → {𝑥𝑥 = (𝐹𝑖)} = {𝑥𝑥 = (𝐹𝑋)})
6 fveq2 6663 . . . . . . . 8 (𝑖 = 𝑌 → (𝐹𝑖) = (𝐹𝑌))
76eqeq2d 2829 . . . . . . 7 (𝑖 = 𝑌 → (𝑥 = (𝐹𝑖) ↔ 𝑥 = (𝐹𝑌)))
87abbidv 2882 . . . . . 6 (𝑖 = 𝑌 → {𝑥𝑥 = (𝐹𝑖)} = {𝑥𝑥 = (𝐹𝑌)})
95, 8iunxprg 5009 . . . . 5 ((𝑋𝑉𝑌𝑊) → 𝑖 ∈ {𝑋, 𝑌} {𝑥𝑥 = (𝐹𝑖)} = ({𝑥𝑥 = (𝐹𝑋)} ∪ {𝑥𝑥 = (𝐹𝑌)}))
109adantr 481 . . . 4 (((𝑋𝑉𝑌𝑊) ∧ 𝐹 Fn {𝑋, 𝑌}) → 𝑖 ∈ {𝑋, 𝑌} {𝑥𝑥 = (𝐹𝑖)} = ({𝑥𝑥 = (𝐹𝑋)} ∪ {𝑥𝑥 = (𝐹𝑌)}))
11 iunab 4966 . . . 4 𝑖 ∈ {𝑋, 𝑌} {𝑥𝑥 = (𝐹𝑖)} = {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹𝑖)}
12 df-sn 4558 . . . . . . 7 {(𝐹𝑋)} = {𝑥𝑥 = (𝐹𝑋)}
1312eqcomi 2827 . . . . . 6 {𝑥𝑥 = (𝐹𝑋)} = {(𝐹𝑋)}
14 df-sn 4558 . . . . . . 7 {(𝐹𝑌)} = {𝑥𝑥 = (𝐹𝑌)}
1514eqcomi 2827 . . . . . 6 {𝑥𝑥 = (𝐹𝑌)} = {(𝐹𝑌)}
1613, 15uneq12i 4134 . . . . 5 ({𝑥𝑥 = (𝐹𝑋)} ∪ {𝑥𝑥 = (𝐹𝑌)}) = ({(𝐹𝑋)} ∪ {(𝐹𝑌)})
17 df-pr 4560 . . . . 5 {(𝐹𝑋), (𝐹𝑌)} = ({(𝐹𝑋)} ∪ {(𝐹𝑌)})
1816, 17eqtr4i 2844 . . . 4 ({𝑥𝑥 = (𝐹𝑋)} ∪ {𝑥𝑥 = (𝐹𝑌)}) = {(𝐹𝑋), (𝐹𝑌)}
1910, 11, 183eqtr3g 2876 . . 3 (((𝑋𝑉𝑌𝑊) ∧ 𝐹 Fn {𝑋, 𝑌}) → {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹𝑖)} = {(𝐹𝑋), (𝐹𝑌)})
202, 19eqtrd 2853 . 2 (((𝑋𝑉𝑌𝑊) ∧ 𝐹 Fn {𝑋, 𝑌}) → ran 𝐹 = {(𝐹𝑋), (𝐹𝑌)})
2120ex 413 1 ((𝑋𝑉𝑌𝑊) → (𝐹 Fn {𝑋, 𝑌} → ran 𝐹 = {(𝐹𝑋), (𝐹𝑌)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  {cab 2796  wrex 3136  cun 3931  {csn 4557  {cpr 4559   ciun 4910  ran crn 5549   Fn wfn 6343  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356
This theorem is referenced by:  imarnf1pr  43358
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