Theorem rnfdmpr 47293

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	⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝐹 Fn {𝑋, 𝑌} → ran 𝐹 = {(𝐹‘𝑋), (𝐹‘𝑌)}))

Proof of Theorem rnfdmpr

Dummy variables 𝑥 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnrnfv 6968	. . . 4 ⊢ (𝐹 Fn {𝑋, 𝑌} → ran 𝐹 = {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹‘𝑖)})
2	1	adantl 481	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ 𝐹 Fn {𝑋, 𝑌}) → ran 𝐹 = {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹‘𝑖)})
3		fveq2 6906	. . . . . . . 8 ⊢ (𝑖 = 𝑋 → (𝐹‘𝑖) = (𝐹‘𝑋))
4	3	eqeq2d 2748	. . . . . . 7 ⊢ (𝑖 = 𝑋 → (𝑥 = (𝐹‘𝑖) ↔ 𝑥 = (𝐹‘𝑋)))
5	4	abbidv 2808	. . . . . 6 ⊢ (𝑖 = 𝑋 → {𝑥 ∣ 𝑥 = (𝐹‘𝑖)} = {𝑥 ∣ 𝑥 = (𝐹‘𝑋)})
6		fveq2 6906	. . . . . . . 8 ⊢ (𝑖 = 𝑌 → (𝐹‘𝑖) = (𝐹‘𝑌))
7	6	eqeq2d 2748	. . . . . . 7 ⊢ (𝑖 = 𝑌 → (𝑥 = (𝐹‘𝑖) ↔ 𝑥 = (𝐹‘𝑌)))
8	7	abbidv 2808	. . . . . 6 ⊢ (𝑖 = 𝑌 → {𝑥 ∣ 𝑥 = (𝐹‘𝑖)} = {𝑥 ∣ 𝑥 = (𝐹‘𝑌)})
9	5, 8	iunxprg 5096	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ∪ 𝑖 ∈ {𝑋, 𝑌} {𝑥 ∣ 𝑥 = (𝐹‘𝑖)} = ({𝑥 ∣ 𝑥 = (𝐹‘𝑋)} ∪ {𝑥 ∣ 𝑥 = (𝐹‘𝑌)}))
10	9	adantr 480	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ 𝐹 Fn {𝑋, 𝑌}) → ∪ 𝑖 ∈ {𝑋, 𝑌} {𝑥 ∣ 𝑥 = (𝐹‘𝑖)} = ({𝑥 ∣ 𝑥 = (𝐹‘𝑋)} ∪ {𝑥 ∣ 𝑥 = (𝐹‘𝑌)}))
11		iunab 5051	. . . 4 ⊢ ∪ 𝑖 ∈ {𝑋, 𝑌} {𝑥 ∣ 𝑥 = (𝐹‘𝑖)} = {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹‘𝑖)}
12		df-sn 4627	. . . . . . 7 ⊢ {(𝐹‘𝑋)} = {𝑥 ∣ 𝑥 = (𝐹‘𝑋)}
13	12	eqcomi 2746	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 = (𝐹‘𝑋)} = {(𝐹‘𝑋)}
14		df-sn 4627	. . . . . . 7 ⊢ {(𝐹‘𝑌)} = {𝑥 ∣ 𝑥 = (𝐹‘𝑌)}
15	14	eqcomi 2746	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 = (𝐹‘𝑌)} = {(𝐹‘𝑌)}
16	13, 15	uneq12i 4166	. . . . 5 ⊢ ({𝑥 ∣ 𝑥 = (𝐹‘𝑋)} ∪ {𝑥 ∣ 𝑥 = (𝐹‘𝑌)}) = ({(𝐹‘𝑋)} ∪ {(𝐹‘𝑌)})
17		df-pr 4629	. . . . 5 ⊢ {(𝐹‘𝑋), (𝐹‘𝑌)} = ({(𝐹‘𝑋)} ∪ {(𝐹‘𝑌)})
18	16, 17	eqtr4i 2768	. . . 4 ⊢ ({𝑥 ∣ 𝑥 = (𝐹‘𝑋)} ∪ {𝑥 ∣ 𝑥 = (𝐹‘𝑌)}) = {(𝐹‘𝑋), (𝐹‘𝑌)}
19	10, 11, 18	3eqtr3g 2800	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ 𝐹 Fn {𝑋, 𝑌}) → {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹‘𝑖)} = {(𝐹‘𝑋), (𝐹‘𝑌)})
20	2, 19	eqtrd 2777	. 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ 𝐹 Fn {𝑋, 𝑌}) → ran 𝐹 = {(𝐹‘𝑋), (𝐹‘𝑌)})
21	20	ex 412	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝐹 Fn {𝑋, 𝑌} → ran 𝐹 = {(𝐹‘𝑋), (𝐹‘𝑌)}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2714  ∃wrex 3070   ∪ cun 3949  {csn 4626  {cpr 4628  ∪ ciun 4991  ran crn 5686   Fn wfn 6556  ‘cfv 6561

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569

This theorem is referenced by:  imarnf1pr  47294