Theorem fimacnvinrn2 6672

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 4086	. . . 4 ⊢ ((𝐴 ∩ 𝐵) ∩ ran 𝐹) = (𝐴 ∩ (𝐵 ∩ ran 𝐹))
2		sseqin2 4082	. . . . . . 7 ⊢ (ran 𝐹 ⊆ 𝐵 ↔ (𝐵 ∩ ran 𝐹) = ran 𝐹)
3	2	biimpi 208	. . . . . 6 ⊢ (ran 𝐹 ⊆ 𝐵 → (𝐵 ∩ ran 𝐹) = ran 𝐹)
4	3	adantl 474	. . . . 5 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (𝐵 ∩ ran 𝐹) = ran 𝐹)
5	4	ineq2d 4079	. . . 4 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (𝐴 ∩ (𝐵 ∩ ran 𝐹)) = (𝐴 ∩ ran 𝐹))
6	1, 5	syl5eq 2828	. . 3 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → ((𝐴 ∩ 𝐵) ∩ ran 𝐹) = (𝐴 ∩ ran 𝐹))
7	6	imaeq2d 5775	. 2 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (◡𝐹 “ ((𝐴 ∩ 𝐵) ∩ ran 𝐹)) = (◡𝐹 “ (𝐴 ∩ ran 𝐹)))
8		fimacnvinrn 6671	. . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ 𝐵)) = (◡𝐹 “ ((𝐴 ∩ 𝐵) ∩ ran 𝐹)))
9	8	adantr 473	. 2 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (◡𝐹 “ (𝐴 ∩ 𝐵)) = (◡𝐹 “ ((𝐴 ∩ 𝐵) ∩ ran 𝐹)))
10		fimacnvinrn 6671	. . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ ran 𝐹)))
11	10	adantr 473	. 2 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ ran 𝐹)))
12	7, 9, 11	3eqtr4rd 2827	1 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1508   ∩ cin 3830   ⊆ wss 3831  ^◡ccnv 5410  ran crn 5412   “ cima 5414  Fun wfun 6187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2752  ax-sep 5064  ax-nul 5071  ax-pr 5190

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2551  df-eu 2589  df-clab 2761  df-cleq 2773  df-clel 2848  df-nfc 2920  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3419  df-sbc 3684  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4182  df-if 4354  df-sn 4445  df-pr 4447  df-op 4451  df-uni 4718  df-br 4935  df-opab 4997  df-id 5316  df-xp 5417  df-rel 5418  df-cnv 5419  df-co 5420  df-dm 5421  df-rn 5422  df-res 5423  df-ima 5424  df-iota 6157  df-fun 6195  df-fn 6196  df-f 6197  df-fo 6199  df-fv 6201

This theorem is referenced by:  eulerpartgbij  31307  orvcval4  31396  preimaioomnf  42463