Theorem rnfdmpr 45503

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 6902	. . . 4 ⊢ (𝐹 Fn {𝑋, 𝑌} → ran 𝐹 = {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹‘𝑖)})
2	1	adantl 482	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ 𝐹 Fn {𝑋, 𝑌}) → ran 𝐹 = {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹‘𝑖)})
3		fveq2 6842	. . . . . . . 8 ⊢ (𝑖 = 𝑋 → (𝐹‘𝑖) = (𝐹‘𝑋))
4	3	eqeq2d 2747	. . . . . . 7 ⊢ (𝑖 = 𝑋 → (𝑥 = (𝐹‘𝑖) ↔ 𝑥 = (𝐹‘𝑋)))
5	4	abbidv 2805	. . . . . 6 ⊢ (𝑖 = 𝑋 → {𝑥 ∣ 𝑥 = (𝐹‘𝑖)} = {𝑥 ∣ 𝑥 = (𝐹‘𝑋)})
6		fveq2 6842	. . . . . . . 8 ⊢ (𝑖 = 𝑌 → (𝐹‘𝑖) = (𝐹‘𝑌))
7	6	eqeq2d 2747	. . . . . . 7 ⊢ (𝑖 = 𝑌 → (𝑥 = (𝐹‘𝑖) ↔ 𝑥 = (𝐹‘𝑌)))
8	7	abbidv 2805	. . . . . 6 ⊢ (𝑖 = 𝑌 → {𝑥 ∣ 𝑥 = (𝐹‘𝑖)} = {𝑥 ∣ 𝑥 = (𝐹‘𝑌)})
9	5, 8	iunxprg 5056	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ∪ 𝑖 ∈ {𝑋, 𝑌} {𝑥 ∣ 𝑥 = (𝐹‘𝑖)} = ({𝑥 ∣ 𝑥 = (𝐹‘𝑋)} ∪ {𝑥 ∣ 𝑥 = (𝐹‘𝑌)}))
10	9	adantr 481	. . . 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 5011	. . . 4 ⊢ ∪ 𝑖 ∈ {𝑋, 𝑌} {𝑥 ∣ 𝑥 = (𝐹‘𝑖)} = {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹‘𝑖)}
12		df-sn 4587	. . . . . . 7 ⊢ {(𝐹‘𝑋)} = {𝑥 ∣ 𝑥 = (𝐹‘𝑋)}
13	12	eqcomi 2745	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 = (𝐹‘𝑋)} = {(𝐹‘𝑋)}
14		df-sn 4587	. . . . . . 7 ⊢ {(𝐹‘𝑌)} = {𝑥 ∣ 𝑥 = (𝐹‘𝑌)}
15	14	eqcomi 2745	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 = (𝐹‘𝑌)} = {(𝐹‘𝑌)}
16	13, 15	uneq12i 4121	. . . . 5 ⊢ ({𝑥 ∣ 𝑥 = (𝐹‘𝑋)} ∪ {𝑥 ∣ 𝑥 = (𝐹‘𝑌)}) = ({(𝐹‘𝑋)} ∪ {(𝐹‘𝑌)})
17		df-pr 4589	. . . . 5 ⊢ {(𝐹‘𝑋), (𝐹‘𝑌)} = ({(𝐹‘𝑋)} ∪ {(𝐹‘𝑌)})
18	16, 17	eqtr4i 2767	. . . 4 ⊢ ({𝑥 ∣ 𝑥 = (𝐹‘𝑋)} ∪ {𝑥 ∣ 𝑥 = (𝐹‘𝑌)}) = {(𝐹‘𝑋), (𝐹‘𝑌)}
19	10, 11, 18	3eqtr3g 2799	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ 𝐹 Fn {𝑋, 𝑌}) → {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹‘𝑖)} = {(𝐹‘𝑋), (𝐹‘𝑌)})
20	2, 19	eqtrd 2776	. 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ 𝐹 Fn {𝑋, 𝑌}) → ran 𝐹 = {(𝐹‘𝑋), (𝐹‘𝑌)})
21	20	ex 413	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝐹 Fn {𝑋, 𝑌} → ran 𝐹 = {(𝐹‘𝑋), (𝐹‘𝑌)}))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2713  ∃wrex 3073   ∪ cun 3908  {csn 4586  {cpr 4588  ∪ ciun 4954  ran crn 5634   Fn wfn 6491  ‘cfv 6496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fn 6499  df-fv 6504

This theorem is referenced by:  imarnf1pr  45504