Theorem rnfdmpr 48490

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 487	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ 𝐹 Fn {𝑋, 𝑌}) → ran 𝐹 = {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹‘𝑖)})
3		fveq2 6893	. . . . . . . 8 ⊢ (𝑖 = 𝑋 → (𝐹‘𝑖) = (𝐹‘𝑋))
4	3	eqeq2d 2774	. . . . . . 7 ⊢ (𝑖 = 𝑋 → (𝑥 = (𝐹‘𝑖) ↔ 𝑥 = (𝐹‘𝑋)))
5	4	abbidv 2829	. . . . . 6 ⊢ (𝑖 = 𝑋 → {𝑥 ∣ 𝑥 = (𝐹‘𝑖)} = {𝑥 ∣ 𝑥 = (𝐹‘𝑋)})
6		fveq2 6893	. . . . . . . 8 ⊢ (𝑖 = 𝑌 → (𝐹‘𝑖) = (𝐹‘𝑌))
7	6	eqeq2d 2774	. . . . . . 7 ⊢ (𝑖 = 𝑌 → (𝑥 = (𝐹‘𝑖) ↔ 𝑥 = (𝐹‘𝑌)))
8	7	abbidv 2829	. . . . . 6 ⊢ (𝑖 = 𝑌 → {𝑥 ∣ 𝑥 = (𝐹‘𝑖)} = {𝑥 ∣ 𝑥 = (𝐹‘𝑌)})
9	5, 8	iunxprg 5058	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ∪ 𝑖 ∈ {𝑋, 𝑌} {𝑥 ∣ 𝑥 = (𝐹‘𝑖)} = ({𝑥 ∣ 𝑥 = (𝐹‘𝑋)} ∪ {𝑥 ∣ 𝑥 = (𝐹‘𝑌)}))
10	9	adantr 486	. . . 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 5012	. . . 4 ⊢ ∪ 𝑖 ∈ {𝑋, 𝑌} {𝑥 ∣ 𝑥 = (𝐹‘𝑖)} = {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹‘𝑖)}
12		df-sn 4588	. . . . . . 7 ⊢ {(𝐹‘𝑋)} = {𝑥 ∣ 𝑥 = (𝐹‘𝑋)}
13	12	eqcomi 2772	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 = (𝐹‘𝑋)} = {(𝐹‘𝑋)}
14		df-sn 4588	. . . . . . 7 ⊢ {(𝐹‘𝑌)} = {𝑥 ∣ 𝑥 = (𝐹‘𝑌)}
15	14	eqcomi 2772	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 = (𝐹‘𝑌)} = {(𝐹‘𝑌)}
16	13, 15	uneq12i 4167	. . . . 5 ⊢ ({𝑥 ∣ 𝑥 = (𝐹‘𝑋)} ∪ {𝑥 ∣ 𝑥 = (𝐹‘𝑌)}) = ({(𝐹‘𝑋)} ∪ {(𝐹‘𝑌)})
17		df-pr 4590	. . . . 5 ⊢ {(𝐹‘𝑋), (𝐹‘𝑌)} = ({(𝐹‘𝑋)} ∪ {(𝐹‘𝑌)})
18	16, 17	eqtr4i 2789	. . . 4 ⊢ ({𝑥 ∣ 𝑥 = (𝐹‘𝑋)} ∪ {𝑥 ∣ 𝑥 = (𝐹‘𝑌)}) = {(𝐹‘𝑋), (𝐹‘𝑌)}
19	10, 11, 18	3eqtr3g 2821	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ 𝐹 Fn {𝑋, 𝑌}) → {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹‘𝑖)} = {(𝐹‘𝑋), (𝐹‘𝑌)})
20	2, 19	eqtrd 2798	. 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ 𝐹 Fn {𝑋, 𝑌}) → ran 𝐹 = {(𝐹‘𝑋), (𝐹‘𝑌)})
21	20	ex 418	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝐹 Fn {𝑋, 𝑌} → ran 𝐹 = {(𝐹‘𝑋), (𝐹‘𝑌)}))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146  {cab 2741  ∃wrex 3089   ∪ cun 4151  {csn 4587  {cpr 4589  ∪ ciun 4953  ran crn 5655   Fn wfn 6541  ‘cfv 6546

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6500  df-fun 6548  df-fn 6549  df-fv 6554

This theorem is used by:  imarnf1pr  48491