Theorem fimacnvinrn2 6512

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 3966	. . . 4 ⊢ ((𝐴 ∩ 𝐵) ∩ ran 𝐹) = (𝐴 ∩ (𝐵 ∩ ran 𝐹))
2		sseqin2 3960	. . . . . . 7 ⊢ (ran 𝐹 ⊆ 𝐵 ↔ (𝐵 ∩ ran 𝐹) = ran 𝐹)
3	2	biimpi 206	. . . . . 6 ⊢ (ran 𝐹 ⊆ 𝐵 → (𝐵 ∩ ran 𝐹) = ran 𝐹)
4	3	adantl 473	. . . . 5 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (𝐵 ∩ ran 𝐹) = ran 𝐹)
5	4	ineq2d 3957	. . . 4 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (𝐴 ∩ (𝐵 ∩ ran 𝐹)) = (𝐴 ∩ ran 𝐹))
6	1, 5	syl5eq 2806	. . 3 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → ((𝐴 ∩ 𝐵) ∩ ran 𝐹) = (𝐴 ∩ ran 𝐹))
7	6	imaeq2d 5624	. 2 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (◡𝐹 “ ((𝐴 ∩ 𝐵) ∩ ran 𝐹)) = (◡𝐹 “ (𝐴 ∩ ran 𝐹)))
8		fimacnvinrn 6511	. . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ 𝐵)) = (◡𝐹 “ ((𝐴 ∩ 𝐵) ∩ ran 𝐹)))
9	8	adantr 472	. 2 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (◡𝐹 “ (𝐴 ∩ 𝐵)) = (◡𝐹 “ ((𝐴 ∩ 𝐵) ∩ ran 𝐹)))
10		fimacnvinrn 6511	. . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ ran 𝐹)))
11	10	adantr 472	. 2 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ ran 𝐹)))
12	7, 9, 11	3eqtr4rd 2805	1 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∩ cin 3714   ⊆ wss 3715  ^◡ccnv 5265  ran crn 5267   “ cima 5269  Fun wfun 6043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fo 6055  df-fv 6057

This theorem is referenced by:  eulerpartgbij  30743  orvcval4  30831  preimaioomnf  41435