Theorem rnfdmpr 47196

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 6981	. . . 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 6920	. . . . . . . 8 ⊢ (𝑖 = 𝑋 → (𝐹‘𝑖) = (𝐹‘𝑋))
4	3	eqeq2d 2751	. . . . . . 7 ⊢ (𝑖 = 𝑋 → (𝑥 = (𝐹‘𝑖) ↔ 𝑥 = (𝐹‘𝑋)))
5	4	abbidv 2811	. . . . . 6 ⊢ (𝑖 = 𝑋 → {𝑥 ∣ 𝑥 = (𝐹‘𝑖)} = {𝑥 ∣ 𝑥 = (𝐹‘𝑋)})
6		fveq2 6920	. . . . . . . 8 ⊢ (𝑖 = 𝑌 → (𝐹‘𝑖) = (𝐹‘𝑌))
7	6	eqeq2d 2751	. . . . . . 7 ⊢ (𝑖 = 𝑌 → (𝑥 = (𝐹‘𝑖) ↔ 𝑥 = (𝐹‘𝑌)))
8	7	abbidv 2811	. . . . . 6 ⊢ (𝑖 = 𝑌 → {𝑥 ∣ 𝑥 = (𝐹‘𝑖)} = {𝑥 ∣ 𝑥 = (𝐹‘𝑌)})
9	5, 8	iunxprg 5119	. . . . 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 5074	. . . 4 ⊢ ∪ 𝑖 ∈ {𝑋, 𝑌} {𝑥 ∣ 𝑥 = (𝐹‘𝑖)} = {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹‘𝑖)}
12		df-sn 4649	. . . . . . 7 ⊢ {(𝐹‘𝑋)} = {𝑥 ∣ 𝑥 = (𝐹‘𝑋)}
13	12	eqcomi 2749	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 = (𝐹‘𝑋)} = {(𝐹‘𝑋)}
14		df-sn 4649	. . . . . . 7 ⊢ {(𝐹‘𝑌)} = {𝑥 ∣ 𝑥 = (𝐹‘𝑌)}
15	14	eqcomi 2749	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 = (𝐹‘𝑌)} = {(𝐹‘𝑌)}
16	13, 15	uneq12i 4189	. . . . 5 ⊢ ({𝑥 ∣ 𝑥 = (𝐹‘𝑋)} ∪ {𝑥 ∣ 𝑥 = (𝐹‘𝑌)}) = ({(𝐹‘𝑋)} ∪ {(𝐹‘𝑌)})
17		df-pr 4651	. . . . 5 ⊢ {(𝐹‘𝑋), (𝐹‘𝑌)} = ({(𝐹‘𝑋)} ∪ {(𝐹‘𝑌)})
18	16, 17	eqtr4i 2771	. . . 4 ⊢ ({𝑥 ∣ 𝑥 = (𝐹‘𝑋)} ∪ {𝑥 ∣ 𝑥 = (𝐹‘𝑌)}) = {(𝐹‘𝑋), (𝐹‘𝑌)}
19	10, 11, 18	3eqtr3g 2803	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ 𝐹 Fn {𝑋, 𝑌}) → {𝑥 ∣ ∃𝑖 ∈ {𝑋, 𝑌}𝑥 = (𝐹‘𝑖)} = {(𝐹‘𝑋), (𝐹‘𝑌)})
20	2, 19	eqtrd 2780	. 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 1537   ∈ wcel 2108  {cab 2717  ∃wrex 3076   ∪ cun 3974  {csn 4648  {cpr 4650  ∪ ciun 5015  ran crn 5701   Fn wfn 6568  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-fv 6581

This theorem is referenced by:  imarnf1pr  47197