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Theorem resima2OLD 5468
 Description: Obsolete proof of resima2 5467 as of 25-Aug-2021. (Contributed by FL, 31-Aug-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
resima2OLD (𝐵𝐶 → ((𝐴𝐶) “ 𝐵) = (𝐴𝐵))

Proof of Theorem resima2OLD
StepHypRef Expression
1 df-ima 5156 . 2 ((𝐴𝐶) “ 𝐵) = ran ((𝐴𝐶) ↾ 𝐵)
2 resres 5444 . . . 4 ((𝐴𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶𝐵))
32rneqi 5384 . . 3 ran ((𝐴𝐶) ↾ 𝐵) = ran (𝐴 ↾ (𝐶𝐵))
4 df-ss 3621 . . . 4 (𝐵𝐶 ↔ (𝐵𝐶) = 𝐵)
5 incom 3838 . . . . . . . 8 (𝐶𝐵) = (𝐵𝐶)
65a1i 11 . . . . . . 7 ((𝐵𝐶) = 𝐵 → (𝐶𝐵) = (𝐵𝐶))
76reseq2d 5428 . . . . . 6 ((𝐵𝐶) = 𝐵 → (𝐴 ↾ (𝐶𝐵)) = (𝐴 ↾ (𝐵𝐶)))
87rneqd 5385 . . . . 5 ((𝐵𝐶) = 𝐵 → ran (𝐴 ↾ (𝐶𝐵)) = ran (𝐴 ↾ (𝐵𝐶)))
9 reseq2 5423 . . . . . . 7 ((𝐵𝐶) = 𝐵 → (𝐴 ↾ (𝐵𝐶)) = (𝐴𝐵))
109rneqd 5385 . . . . . 6 ((𝐵𝐶) = 𝐵 → ran (𝐴 ↾ (𝐵𝐶)) = ran (𝐴𝐵))
11 df-ima 5156 . . . . . 6 (𝐴𝐵) = ran (𝐴𝐵)
1210, 11syl6eqr 2703 . . . . 5 ((𝐵𝐶) = 𝐵 → ran (𝐴 ↾ (𝐵𝐶)) = (𝐴𝐵))
138, 12eqtrd 2685 . . . 4 ((𝐵𝐶) = 𝐵 → ran (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
144, 13sylbi 207 . . 3 (𝐵𝐶 → ran (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
153, 14syl5eq 2697 . 2 (𝐵𝐶 → ran ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))
161, 15syl5eq 2697 1 (𝐵𝐶 → ((𝐴𝐶) “ 𝐵) = (𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1523   ∩ cin 3606   ⊆ wss 3607  ran crn 5144   ↾ cres 5145   “ cima 5146 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156 This theorem is referenced by: (None)
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