Theorem imarnf1pr 47276

Description: The image of the range of a function 𝐹 under a function 𝐸 if 𝐹 is a function from a pair into the domain of 𝐸. (Contributed by Alexander van der Vekens, 2-Feb-2018.)

Assertion

Ref	Expression
imarnf1pr	⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ ((𝐸‘(𝐹‘𝑋)) = 𝐴 ∧ (𝐸‘(𝐹‘𝑌)) = 𝐵)) → (𝐸 “ ran 𝐹) = {𝐴, 𝐵}))

Proof of Theorem imarnf1pr

Step	Hyp	Ref	Expression
1		ffn 6652	. . . . . . . . 9 ⊢ (𝐸:dom 𝐸⟶𝑅 → 𝐸 Fn dom 𝐸)
2	1	adantl 481	. . . . . . . 8 ⊢ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) → 𝐸 Fn dom 𝐸)
3	2	adantr 480	. . . . . . 7 ⊢ (((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊)) → 𝐸 Fn dom 𝐸)
4		simpll 766	. . . . . . . 8 ⊢ (((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊)) → 𝐹:{𝑋, 𝑌}⟶dom 𝐸)
5		prid1g 4712	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋, 𝑌})
6	5	adantr 480	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → 𝑋 ∈ {𝑋, 𝑌})
7	6	adantl 481	. . . . . . . 8 ⊢ (((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊)) → 𝑋 ∈ {𝑋, 𝑌})
8	4, 7	ffvelcdmd 7019	. . . . . . 7 ⊢ (((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊)) → (𝐹‘𝑋) ∈ dom 𝐸)
9		prid2g 4713	. . . . . . . . 9 ⊢ (𝑌 ∈ 𝑊 → 𝑌 ∈ {𝑋, 𝑌})
10	9	ad2antll 729	. . . . . . . 8 ⊢ (((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊)) → 𝑌 ∈ {𝑋, 𝑌})
11	4, 10	ffvelcdmd 7019	. . . . . . 7 ⊢ (((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊)) → (𝐹‘𝑌) ∈ dom 𝐸)
12		fnimapr 6906	. . . . . . 7 ⊢ ((𝐸 Fn dom 𝐸 ∧ (𝐹‘𝑋) ∈ dom 𝐸 ∧ (𝐹‘𝑌) ∈ dom 𝐸) → (𝐸 “ {(𝐹‘𝑋), (𝐹‘𝑌)}) = {(𝐸‘(𝐹‘𝑋)), (𝐸‘(𝐹‘𝑌))})
13	3, 8, 11, 12	syl3anc 1373	. . . . . 6 ⊢ (((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊)) → (𝐸 “ {(𝐹‘𝑋), (𝐹‘𝑌)}) = {(𝐸‘(𝐹‘𝑋)), (𝐸‘(𝐹‘𝑌))})
14	13	ex 412	. . . . 5 ⊢ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝐸 “ {(𝐹‘𝑋), (𝐹‘𝑌)}) = {(𝐸‘(𝐹‘𝑋)), (𝐸‘(𝐹‘𝑌))}))
15	14	adantr 480	. . . 4 ⊢ (((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ ((𝐸‘(𝐹‘𝑋)) = 𝐴 ∧ (𝐸‘(𝐹‘𝑌)) = 𝐵)) → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝐸 “ {(𝐹‘𝑋), (𝐹‘𝑌)}) = {(𝐸‘(𝐹‘𝑋)), (𝐸‘(𝐹‘𝑌))}))
16	15	impcom 407	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ ((𝐸‘(𝐹‘𝑋)) = 𝐴 ∧ (𝐸‘(𝐹‘𝑌)) = 𝐵))) → (𝐸 “ {(𝐹‘𝑋), (𝐹‘𝑌)}) = {(𝐸‘(𝐹‘𝑋)), (𝐸‘(𝐹‘𝑌))})
17		ffn 6652	. . . . . . . . 9 ⊢ (𝐹:{𝑋, 𝑌}⟶dom 𝐸 → 𝐹 Fn {𝑋, 𝑌})
18		rnfdmpr 47275	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝐹 Fn {𝑋, 𝑌} → ran 𝐹 = {(𝐹‘𝑋), (𝐹‘𝑌)}))
19	17, 18	syl5com 31	. . . . . . . 8 ⊢ (𝐹:{𝑋, 𝑌}⟶dom 𝐸 → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ran 𝐹 = {(𝐹‘𝑋), (𝐹‘𝑌)}))
20	19	adantr 480	. . . . . . 7 ⊢ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ran 𝐹 = {(𝐹‘𝑋), (𝐹‘𝑌)}))
21	20	adantr 480	. . . . . 6 ⊢ (((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ ((𝐸‘(𝐹‘𝑋)) = 𝐴 ∧ (𝐸‘(𝐹‘𝑌)) = 𝐵)) → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ran 𝐹 = {(𝐹‘𝑋), (𝐹‘𝑌)}))
22	21	impcom 407	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ ((𝐸‘(𝐹‘𝑋)) = 𝐴 ∧ (𝐸‘(𝐹‘𝑌)) = 𝐵))) → ran 𝐹 = {(𝐹‘𝑋), (𝐹‘𝑌)})
23	22	eqcomd 2735	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ ((𝐸‘(𝐹‘𝑋)) = 𝐴 ∧ (𝐸‘(𝐹‘𝑌)) = 𝐵))) → {(𝐹‘𝑋), (𝐹‘𝑌)} = ran 𝐹)
24	23	imaeq2d 6011	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ ((𝐸‘(𝐹‘𝑋)) = 𝐴 ∧ (𝐸‘(𝐹‘𝑌)) = 𝐵))) → (𝐸 “ {(𝐹‘𝑋), (𝐹‘𝑌)}) = (𝐸 “ ran 𝐹))
25		preq12 4687	. . . 4 ⊢ (((𝐸‘(𝐹‘𝑋)) = 𝐴 ∧ (𝐸‘(𝐹‘𝑌)) = 𝐵) → {(𝐸‘(𝐹‘𝑋)), (𝐸‘(𝐹‘𝑌))} = {𝐴, 𝐵})
26	25	ad2antll 729	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ ((𝐸‘(𝐹‘𝑋)) = 𝐴 ∧ (𝐸‘(𝐹‘𝑌)) = 𝐵))) → {(𝐸‘(𝐹‘𝑋)), (𝐸‘(𝐹‘𝑌))} = {𝐴, 𝐵})
27	16, 24, 26	3eqtr3d 2772	. 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ ((𝐸‘(𝐹‘𝑋)) = 𝐴 ∧ (𝐸‘(𝐹‘𝑌)) = 𝐵))) → (𝐸 “ ran 𝐹) = {𝐴, 𝐵})
28	27	ex 412	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ ((𝐸‘(𝐹‘𝑋)) = 𝐴 ∧ (𝐸‘(𝐹‘𝑌)) = 𝐵)) → (𝐸 “ ran 𝐹) = {𝐴, 𝐵}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cpr 4579  dom cdm 5619  ran crn 5620   “ cima 5622   Fn wfn 6477  ⟶wf 6478  ‘cfv 6482

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-fv 6490

This theorem is referenced by: (None)