Theorem k0004lem1 44109

Description: Application of ssin 4260 to range of a function. (Contributed by RP, 1-Apr-2021.)

Assertion

Ref	Expression
k0004lem1	⊢ (𝐷 = (𝐵 ∩ 𝐶) → ((𝐹:𝐴⟶𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ 𝐹:𝐴⟶𝐷))

Proof of Theorem k0004lem1

Step	Hyp	Ref	Expression
1		fnima 6710	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)
2	1	sseq1d 4040	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → ((𝐹 “ 𝐴) ⊆ 𝐶 ↔ ran 𝐹 ⊆ 𝐶))
3	2	anbi2d 629	. . . . 5 ⊢ (𝐹 Fn 𝐴 → ((ran 𝐹 ⊆ 𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ (ran 𝐹 ⊆ 𝐵 ∧ ran 𝐹 ⊆ 𝐶)))
4		ssin 4260	. . . . 5 ⊢ ((ran 𝐹 ⊆ 𝐵 ∧ ran 𝐹 ⊆ 𝐶) ↔ ran 𝐹 ⊆ (𝐵 ∩ 𝐶))
5	3, 4	bitrdi 287	. . . 4 ⊢ (𝐹 Fn 𝐴 → ((ran 𝐹 ⊆ 𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ ran 𝐹 ⊆ (𝐵 ∩ 𝐶)))
6	5	pm5.32i 574	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ (ran 𝐹 ⊆ 𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶)) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝐵 ∩ 𝐶)))
7		df-f 6577	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
8	7	anbi1i 623	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ∧ (𝐹 “ 𝐴) ⊆ 𝐶))
9		anass 468	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ (𝐹 Fn 𝐴 ∧ (ran 𝐹 ⊆ 𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶)))
10	8, 9	bitri 275	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ (𝐹 Fn 𝐴 ∧ (ran 𝐹 ⊆ 𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶)))
11		df-f 6577	. . 3 ⊢ (𝐹:𝐴⟶(𝐵 ∩ 𝐶) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝐵 ∩ 𝐶)))
12	6, 10, 11	3bitr4i 303	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ 𝐹:𝐴⟶(𝐵 ∩ 𝐶))
13		feq3 6730	. 2 ⊢ (𝐷 = (𝐵 ∩ 𝐶) → (𝐹:𝐴⟶𝐷 ↔ 𝐹:𝐴⟶(𝐵 ∩ 𝐶)))
14	12, 13	bitr4id 290	1 ⊢ (𝐷 = (𝐵 ∩ 𝐶) → ((𝐹:𝐴⟶𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ 𝐹:𝐴⟶𝐷))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∩ cin 3975   ⊆ wss 3976  ran crn 5701   “ cima 5703   Fn wfn 6568  ⟶wf 6569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-fun 6575  df-fn 6576  df-f 6577

This theorem is referenced by:  k0004lem2  44110