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Theorem rnfdmpr 46287
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 6950 . . . 4 (𝐹 Fn {𝑋, 𝑌} → ran 𝐹 = {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹𝑖)})
21adantl 480 . . 3 (((𝑋𝑉𝑌𝑊) ∧ 𝐹 Fn {𝑋, 𝑌}) → ran 𝐹 = {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹𝑖)})
3 fveq2 6890 . . . . . . . 8 (𝑖 = 𝑋 → (𝐹𝑖) = (𝐹𝑋))
43eqeq2d 2741 . . . . . . 7 (𝑖 = 𝑋 → (𝑥 = (𝐹𝑖) ↔ 𝑥 = (𝐹𝑋)))
54abbidv 2799 . . . . . 6 (𝑖 = 𝑋 → {𝑥𝑥 = (𝐹𝑖)} = {𝑥𝑥 = (𝐹𝑋)})
6 fveq2 6890 . . . . . . . 8 (𝑖 = 𝑌 → (𝐹𝑖) = (𝐹𝑌))
76eqeq2d 2741 . . . . . . 7 (𝑖 = 𝑌 → (𝑥 = (𝐹𝑖) ↔ 𝑥 = (𝐹𝑌)))
87abbidv 2799 . . . . . 6 (𝑖 = 𝑌 → {𝑥𝑥 = (𝐹𝑖)} = {𝑥𝑥 = (𝐹𝑌)})
95, 8iunxprg 5098 . . . . 5 ((𝑋𝑉𝑌𝑊) → 𝑖 ∈ {𝑋, 𝑌} {𝑥𝑥 = (𝐹𝑖)} = ({𝑥𝑥 = (𝐹𝑋)} ∪ {𝑥𝑥 = (𝐹𝑌)}))
109adantr 479 . . . 4 (((𝑋𝑉𝑌𝑊) ∧ 𝐹 Fn {𝑋, 𝑌}) → 𝑖 ∈ {𝑋, 𝑌} {𝑥𝑥 = (𝐹𝑖)} = ({𝑥𝑥 = (𝐹𝑋)} ∪ {𝑥𝑥 = (𝐹𝑌)}))
11 iunab 5053 . . . 4 𝑖 ∈ {𝑋, 𝑌} {𝑥𝑥 = (𝐹𝑖)} = {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹𝑖)}
12 df-sn 4628 . . . . . . 7 {(𝐹𝑋)} = {𝑥𝑥 = (𝐹𝑋)}
1312eqcomi 2739 . . . . . 6 {𝑥𝑥 = (𝐹𝑋)} = {(𝐹𝑋)}
14 df-sn 4628 . . . . . . 7 {(𝐹𝑌)} = {𝑥𝑥 = (𝐹𝑌)}
1514eqcomi 2739 . . . . . 6 {𝑥𝑥 = (𝐹𝑌)} = {(𝐹𝑌)}
1613, 15uneq12i 4160 . . . . 5 ({𝑥𝑥 = (𝐹𝑋)} ∪ {𝑥𝑥 = (𝐹𝑌)}) = ({(𝐹𝑋)} ∪ {(𝐹𝑌)})
17 df-pr 4630 . . . . 5 {(𝐹𝑋), (𝐹𝑌)} = ({(𝐹𝑋)} ∪ {(𝐹𝑌)})
1816, 17eqtr4i 2761 . . . 4 ({𝑥𝑥 = (𝐹𝑋)} ∪ {𝑥𝑥 = (𝐹𝑌)}) = {(𝐹𝑋), (𝐹𝑌)}
1910, 11, 183eqtr3g 2793 . . 3 (((𝑋𝑉𝑌𝑊) ∧ 𝐹 Fn {𝑋, 𝑌}) → {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹𝑖)} = {(𝐹𝑋), (𝐹𝑌)})
202, 19eqtrd 2770 . 2 (((𝑋𝑉𝑌𝑊) ∧ 𝐹 Fn {𝑋, 𝑌}) → ran 𝐹 = {(𝐹𝑋), (𝐹𝑌)})
2120ex 411 1 ((𝑋𝑉𝑌𝑊) → (𝐹 Fn {𝑋, 𝑌} → ran 𝐹 = {(𝐹𝑋), (𝐹𝑌)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wcel 2104  {cab 2707  wrex 3068  cun 3945  {csn 4627  {cpr 4629   ciun 4996  ran crn 5676   Fn wfn 6537  cfv 6542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550
This theorem is referenced by:  imarnf1pr  46288
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