Theorem imarnf1pr 47311

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 6706	. . . . . . . . 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 4736	. . . . . . . . . 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 7075	. . . . . . 7 ⊢ (((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊)) → (𝐹‘𝑋) ∈ dom 𝐸)
9		prid2g 4737	. . . . . . . . 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 7075	. . . . . . 7 ⊢ (((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊)) → (𝐹‘𝑌) ∈ dom 𝐸)
12		fnimapr 6962	. . . . . . 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 6706	. . . . . . . . 9 ⊢ (𝐹:{𝑋, 𝑌}⟶dom 𝐸 → 𝐹 Fn {𝑋, 𝑌})
18		rnfdmpr 47310	. . . . . . . . 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 2741	. . . 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 6047	. . 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 4711	. . . 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 2778	. 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 2108  {cpr 4603  dom cdm 5654  ran crn 5655   “ cima 5657   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539

This theorem is referenced by: (None)