Theorem k0004lem1 44574

Description: Application of ssin 4179 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 6628	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)
2	1	sseq1d 3953	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → ((𝐹 “ 𝐴) ⊆ 𝐶 ↔ ran 𝐹 ⊆ 𝐶))
3	2	anbi2d 631	. . . . 5 ⊢ (𝐹 Fn 𝐴 → ((ran 𝐹 ⊆ 𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ (ran 𝐹 ⊆ 𝐵 ∧ ran 𝐹 ⊆ 𝐶)))
4		ssin 4179	. . . . 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 6502	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
8	7	anbi1i 625	. . . 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 6502	. . 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 6648	. 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 1542   ∩ cin 3888   ⊆ wss 3889  ran crn 5632   “ cima 5634   Fn wfn 6493  ⟶wf 6494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-fun 6500  df-fn 6501  df-f 6502

This theorem is referenced by:  k0004lem2  44575