Theorem rnfdmpr 45989

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 6952	. . . 4 ⊢ (𝐹 Fn {𝑋, 𝑌} → ran 𝐹 = {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹‘𝑖)})
2	1	adantl 483	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ 𝐹 Fn {𝑋, 𝑌}) → ran 𝐹 = {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹‘𝑖)})
3		fveq2 6892	. . . . . . . 8 ⊢ (𝑖 = 𝑋 → (𝐹‘𝑖) = (𝐹‘𝑋))
4	3	eqeq2d 2744	. . . . . . 7 ⊢ (𝑖 = 𝑋 → (𝑥 = (𝐹‘𝑖) ↔ 𝑥 = (𝐹‘𝑋)))
5	4	abbidv 2802	. . . . . 6 ⊢ (𝑖 = 𝑋 → {𝑥 ∣ 𝑥 = (𝐹‘𝑖)} = {𝑥 ∣ 𝑥 = (𝐹‘𝑋)})
6		fveq2 6892	. . . . . . . 8 ⊢ (𝑖 = 𝑌 → (𝐹‘𝑖) = (𝐹‘𝑌))
7	6	eqeq2d 2744	. . . . . . 7 ⊢ (𝑖 = 𝑌 → (𝑥 = (𝐹‘𝑖) ↔ 𝑥 = (𝐹‘𝑌)))
8	7	abbidv 2802	. . . . . 6 ⊢ (𝑖 = 𝑌 → {𝑥 ∣ 𝑥 = (𝐹‘𝑖)} = {𝑥 ∣ 𝑥 = (𝐹‘𝑌)})
9	5, 8	iunxprg 5100	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ∪ 𝑖 ∈ {𝑋, 𝑌} {𝑥 ∣ 𝑥 = (𝐹‘𝑖)} = ({𝑥 ∣ 𝑥 = (𝐹‘𝑋)} ∪ {𝑥 ∣ 𝑥 = (𝐹‘𝑌)}))
10	9	adantr 482	. . . 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 5055	. . . 4 ⊢ ∪ 𝑖 ∈ {𝑋, 𝑌} {𝑥 ∣ 𝑥 = (𝐹‘𝑖)} = {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹‘𝑖)}
12		df-sn 4630	. . . . . . 7 ⊢ {(𝐹‘𝑋)} = {𝑥 ∣ 𝑥 = (𝐹‘𝑋)}
13	12	eqcomi 2742	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 = (𝐹‘𝑋)} = {(𝐹‘𝑋)}
14		df-sn 4630	. . . . . . 7 ⊢ {(𝐹‘𝑌)} = {𝑥 ∣ 𝑥 = (𝐹‘𝑌)}
15	14	eqcomi 2742	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 = (𝐹‘𝑌)} = {(𝐹‘𝑌)}
16	13, 15	uneq12i 4162	. . . . 5 ⊢ ({𝑥 ∣ 𝑥 = (𝐹‘𝑋)} ∪ {𝑥 ∣ 𝑥 = (𝐹‘𝑌)}) = ({(𝐹‘𝑋)} ∪ {(𝐹‘𝑌)})
17		df-pr 4632	. . . . 5 ⊢ {(𝐹‘𝑋), (𝐹‘𝑌)} = ({(𝐹‘𝑋)} ∪ {(𝐹‘𝑌)})
18	16, 17	eqtr4i 2764	. . . 4 ⊢ ({𝑥 ∣ 𝑥 = (𝐹‘𝑋)} ∪ {𝑥 ∣ 𝑥 = (𝐹‘𝑌)}) = {(𝐹‘𝑋), (𝐹‘𝑌)}
19	10, 11, 18	3eqtr3g 2796	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ 𝐹 Fn {𝑋, 𝑌}) → {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹‘𝑖)} = {(𝐹‘𝑋), (𝐹‘𝑌)})
20	2, 19	eqtrd 2773	. 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ 𝐹 Fn {𝑋, 𝑌}) → ran 𝐹 = {(𝐹‘𝑋), (𝐹‘𝑌)})
21	20	ex 414	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝐹 Fn {𝑋, 𝑌} → ran 𝐹 = {(𝐹‘𝑋), (𝐹‘𝑌)}))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710  ∃wrex 3071   ∪ cun 3947  {csn 4629  {cpr 4631  ∪ ciun 4998  ran crn 5678   Fn wfn 6539  ‘cfv 6544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552

This theorem is referenced by:  imarnf1pr  45990