Theorem k0004lem1 41739

Description: Application of ssin 4170 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 6561	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)
2	1	sseq1d 3957	. . . . . 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 4170	. . . . 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 575	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ (ran 𝐹 ⊆ 𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶)) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝐵 ∩ 𝐶)))
7		df-f 6436	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
8	7	anbi1i 624	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ∧ (𝐹 “ 𝐴) ⊆ 𝐶))
9		anass 469	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ (𝐹 Fn 𝐴 ∧ (ran 𝐹 ⊆ 𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶)))
10	8, 9	bitri 274	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ (𝐹 Fn 𝐴 ∧ (ran 𝐹 ⊆ 𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶)))
11		df-f 6436	. . 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 6581	. 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 205   ∧ wa 396   = wceq 1542   ∩ cin 3891   ⊆ wss 3892  ran crn 5591   “ cima 5593   Fn wfn 6427  ⟶wf 6428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-opab 5142  df-xp 5596  df-rel 5597  df-cnv 5598  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-fun 6434  df-fn 6435  df-f 6436

This theorem is referenced by:  k0004lem2  41740