Theorem fimacnvinrn2 7072

Description: Taking the converse image of a set can be limited to the range of the function used. (Contributed by Thierry Arnoux, 17-Feb-2017.)

Assertion

Ref	Expression
fimacnvinrn2	⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ 𝐵)))

Proof of Theorem fimacnvinrn2

Step	Hyp	Ref	Expression
1		inass 4124	. . . 4 ⊢ ((𝐴 ∩ 𝐵) ∩ ran 𝐹) = (𝐴 ∩ (𝐵 ∩ ran 𝐹))
2		sseqin2 4120	. . . . . 6 ⊢ (ran 𝐹 ⊆ 𝐵 ↔ (𝐵 ∩ ran 𝐹) = ran 𝐹)
3	2	bilani 510	. . . . 5 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (𝐵 ∩ ran 𝐹) = ran 𝐹)
4	3	ineq2d 4117	. . . 4 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (𝐴 ∩ (𝐵 ∩ ran 𝐹)) = (𝐴 ∩ ran 𝐹))
5	1, 4	eqtrid 2807	. . 3 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → ((𝐴 ∩ 𝐵) ∩ ran 𝐹) = (𝐴 ∩ ran 𝐹))
6	5	imaeq2d 6051	. 2 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (◡𝐹 “ ((𝐴 ∩ 𝐵) ∩ ran 𝐹)) = (◡𝐹 “ (𝐴 ∩ ran 𝐹)))
7		fimacnvinrn 7071	. . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ 𝐵)) = (◡𝐹 “ ((𝐴 ∩ 𝐵) ∩ ran 𝐹)))
8	7	adantr 486	. 2 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (◡𝐹 “ (𝐴 ∩ 𝐵)) = (◡𝐹 “ ((𝐴 ∩ 𝐵) ∩ ran 𝐹)))
9		fimacnvinrn 7071	. . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ ran 𝐹)))
10	9	adantr 486	. 2 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ ran 𝐹)))
11	6, 8, 10	3eqtr4rd 2806	1 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ 𝐵)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ⊆ wss 3895   ∩ cin 4089  ^◡ccnv 5649  ran crn 5651   “ cima 5653  Fun wfun 6532

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544

This theorem is used by:  eulerpartgbij  35004  orvcval4  35093  preimaioomnf  47729