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Theorem rnfdmpr 41808
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 6404 . . . 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 6352 . . . . . . . 8 (𝑖 = 𝑋 → (𝐹𝑖) = (𝐹𝑋))
43eqeq2d 2770 . . . . . . 7 (𝑖 = 𝑋 → (𝑥 = (𝐹𝑖) ↔ 𝑥 = (𝐹𝑋)))
54abbidv 2879 . . . . . 6 (𝑖 = 𝑋 → {𝑥𝑥 = (𝐹𝑖)} = {𝑥𝑥 = (𝐹𝑋)})
6 fveq2 6352 . . . . . . . 8 (𝑖 = 𝑌 → (𝐹𝑖) = (𝐹𝑌))
76eqeq2d 2770 . . . . . . 7 (𝑖 = 𝑌 → (𝑥 = (𝐹𝑖) ↔ 𝑥 = (𝐹𝑌)))
87abbidv 2879 . . . . . 6 (𝑖 = 𝑌 → {𝑥𝑥 = (𝐹𝑖)} = {𝑥𝑥 = (𝐹𝑌)})
95, 8iunxprg 4759 . . . . 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 4718 . . . 4 𝑖 ∈ {𝑋, 𝑌} {𝑥𝑥 = (𝐹𝑖)} = {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹𝑖)}
12 df-sn 4322 . . . . . . 7 {(𝐹𝑋)} = {𝑥𝑥 = (𝐹𝑋)}
1312eqcomi 2769 . . . . . 6 {𝑥𝑥 = (𝐹𝑋)} = {(𝐹𝑋)}
14 df-sn 4322 . . . . . . 7 {(𝐹𝑌)} = {𝑥𝑥 = (𝐹𝑌)}
1514eqcomi 2769 . . . . . 6 {𝑥𝑥 = (𝐹𝑌)} = {(𝐹𝑌)}
1613, 15uneq12i 3908 . . . . 5 ({𝑥𝑥 = (𝐹𝑋)} ∪ {𝑥𝑥 = (𝐹𝑌)}) = ({(𝐹𝑋)} ∪ {(𝐹𝑌)})
17 df-pr 4324 . . . . 5 {(𝐹𝑋), (𝐹𝑌)} = ({(𝐹𝑋)} ∪ {(𝐹𝑌)})
1816, 17eqtr4i 2785 . . . 4 ({𝑥𝑥 = (𝐹𝑋)} ∪ {𝑥𝑥 = (𝐹𝑌)}) = {(𝐹𝑋), (𝐹𝑌)}
1910, 11, 183eqtr3g 2817 . . 3 (((𝑋𝑉𝑌𝑊) ∧ 𝐹 Fn {𝑋, 𝑌}) → {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹𝑖)} = {(𝐹𝑋), (𝐹𝑌)})
202, 19eqtrd 2794 . 2 (((𝑋𝑉𝑌𝑊) ∧ 𝐹 Fn {𝑋, 𝑌}) → ran 𝐹 = {(𝐹𝑋), (𝐹𝑌)})
2120ex 449 1 ((𝑋𝑉𝑌𝑊) → (𝐹 Fn {𝑋, 𝑌} → ran 𝐹 = {(𝐹𝑋), (𝐹𝑌)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  {cab 2746  wrex 3051  cun 3713  {csn 4321  {cpr 4323   ciun 4672  ran crn 5267   Fn wfn 6044  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057
This theorem is referenced by:  imarnf1pr  41809
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