Theorem imarnf1pr 47943

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 486	. . . . . . . 8 ⊢ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) → 𝐸 Fn dom 𝐸)
3	2	adantr 485	. . . . . . 7 ⊢ (((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊)) → 𝐸 Fn dom 𝐸)
4		simpll 778	. . . . . . . 8 ⊢ (((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊)) → 𝐹:{𝑋, 𝑌}⟶dom 𝐸)
5		prid1g 4729	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋, 𝑌})
6	5	adantr 485	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → 𝑋 ∈ {𝑋, 𝑌})
7	6	adantl 486	. . . . . . . 8 ⊢ (((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊)) → 𝑋 ∈ {𝑋, 𝑌})
8	4, 7	ffvelcdmd 7081	. . . . . . 7 ⊢ (((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊)) → (𝐹‘𝑋) ∈ dom 𝐸)
9		prid2g 4730	. . . . . . . . 9 ⊢ (𝑌 ∈ 𝑊 → 𝑌 ∈ {𝑋, 𝑌})
10	9	ad2antll 741	. . . . . . . 8 ⊢ (((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊)) → 𝑌 ∈ {𝑋, 𝑌})
11	4, 10	ffvelcdmd 7081	. . . . . . 7 ⊢ (((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊)) → (𝐹‘𝑌) ∈ dom 𝐸)
12		fnimapr 6965	. . . . . . 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 1396	. . . . . 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 417	. . . . 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 485	. . . 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 412	. . 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 47942	. . . . . . . . 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 485	. . . . . . 7 ⊢ ((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ran 𝐹 = {(𝐹‘𝑋), (𝐹‘𝑌)}))
21	20	adantr 485	. . . . . 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 412	. . . . 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 2775	. . . 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 6063	. . 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 4704	. . . 4 ⊢ (((𝐸‘(𝐹‘𝑋)) = 𝐴 ∧ (𝐸‘(𝐹‘𝑌)) = 𝐵) → {(𝐸‘(𝐹‘𝑋)), (𝐸‘(𝐹‘𝑌))} = {𝐴, 𝐵})
26	25	ad2antll 741	. . 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 2812	. 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 417	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 400   = wceq 1567   ∈ wcel 2149  {cpr 4594  dom cdm 5662  ran crn 5663   “ cima 5665   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5259  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3423  df-v 3463  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545

This theorem is referenced by: (None)