Theorem imarnf1pr 48557

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 6731	. . . . . . . . 9 ⊢ (𝐸:dom 𝐸⟶𝑅 → 𝐸 Fn dom 𝐸)
2	1	adantl 487	. . . . . . . 8 ⊢ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) → 𝐸 Fn dom 𝐸)
3	2	adantr 486	. . . . . . 7 ⊢ (((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊)) → 𝐸 Fn dom 𝐸)
4		simpll 779	. . . . . . . 8 ⊢ (((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊)) → 𝐹:{𝑋, 𝑌}⟶dom 𝐸)
5		prid1g 4721	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋, 𝑌})
6	5	adantr 486	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → 𝑋 ∈ {𝑋, 𝑌})
7	6	adantl 487	. . . . . . . 8 ⊢ (((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊)) → 𝑋 ∈ {𝑋, 𝑌})
8	4, 7	ffvelcdmd 7109	. . . . . . 7 ⊢ (((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊)) → (𝐹‘𝑋) ∈ dom 𝐸)
9		prid2g 4722	. . . . . . . . 9 ⊢ (𝑌 ∈ 𝑊 → 𝑌 ∈ {𝑋, 𝑌})
10	9	ad2antll 742	. . . . . . . 8 ⊢ (((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊)) → 𝑌 ∈ {𝑋, 𝑌})
11	4, 10	ffvelcdmd 7109	. . . . . . 7 ⊢ (((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊)) → (𝐹‘𝑌) ∈ dom 𝐸)
12		fnimapr 6991	. . . . . . 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 1399	. . . . . 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 418	. . . . 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 486	. . . 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 413	. . 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 6731	. . . . . . . . 9 ⊢ (𝐹:{𝑋, 𝑌}⟶dom 𝐸 → 𝐹 Fn {𝑋, 𝑌})
18		rnfdmpr 48556	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝐹 Fn {𝑋, 𝑌} → ran 𝐹 = {(𝐹‘𝑋), (𝐹‘𝑌)}))
19	17, 18	syl5com 32	. . . . . . . 8 ⊢ (𝐹:{𝑋, 𝑌}⟶dom 𝐸 → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ran 𝐹 = {(𝐹‘𝑋), (𝐹‘𝑌)}))
20	19	adantr 486	. . . . . . 7 ⊢ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ran 𝐹 = {(𝐹‘𝑋), (𝐹‘𝑌)}))
21	20	adantr 486	. . . . . 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 413	. . . . 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 2769	. . . 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 6064	. . 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 4696	. . . 4 ⊢ (((𝐸‘(𝐹‘𝑋)) = 𝐴 ∧ (𝐸‘(𝐹‘𝑌)) = 𝐵) → {(𝐸‘(𝐹‘𝑋)), (𝐸‘(𝐹‘𝑌))} = {𝐴, 𝐵})
26	25	ad2antll 742	. . 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 2806	. 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 418	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}))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cpr 4590  dom cdm 5655  ran crn 5656   “ cima 5658   Fn wfn 6553  ⟶wf 6554  ‘cfv 6558

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-fv 6566

This theorem is used by: (None)