Theorem k0004lem1 39415

Description: Application of ssin 4055 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		feq3 6276	. 2 ⊢ (𝐷 = (𝐵 ∩ 𝐶) → (𝐹:𝐴⟶𝐷 ↔ 𝐹:𝐴⟶(𝐵 ∩ 𝐶)))
2		fnima 6258	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)
3	2	sseq1d 3851	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → ((𝐹 “ 𝐴) ⊆ 𝐶 ↔ ran 𝐹 ⊆ 𝐶))
4	3	anbi2d 622	. . . . 5 ⊢ (𝐹 Fn 𝐴 → ((ran 𝐹 ⊆ 𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ (ran 𝐹 ⊆ 𝐵 ∧ ran 𝐹 ⊆ 𝐶)))
5		ssin 4055	. . . . 5 ⊢ ((ran 𝐹 ⊆ 𝐵 ∧ ran 𝐹 ⊆ 𝐶) ↔ ran 𝐹 ⊆ (𝐵 ∩ 𝐶))
6	4, 5	syl6bb 279	. . . 4 ⊢ (𝐹 Fn 𝐴 → ((ran 𝐹 ⊆ 𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ ran 𝐹 ⊆ (𝐵 ∩ 𝐶)))
7	6	pm5.32i 570	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ (ran 𝐹 ⊆ 𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶)) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝐵 ∩ 𝐶)))
8		df-f 6141	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
9	8	anbi1i 617	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ∧ (𝐹 “ 𝐴) ⊆ 𝐶))
10		anass 462	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ (𝐹 Fn 𝐴 ∧ (ran 𝐹 ⊆ 𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶)))
11	9, 10	bitri 267	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ (𝐹 Fn 𝐴 ∧ (ran 𝐹 ⊆ 𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶)))
12		df-f 6141	. . 3 ⊢ (𝐹:𝐴⟶(𝐵 ∩ 𝐶) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝐵 ∩ 𝐶)))
13	7, 11, 12	3bitr4i 295	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ 𝐹:𝐴⟶(𝐵 ∩ 𝐶))
14	1, 13	syl6rbbr 282	1 ⊢ (𝐷 = (𝐵 ∩ 𝐶) → ((𝐹:𝐴⟶𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ 𝐹:𝐴⟶𝐷))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601   ∩ cin 3791   ⊆ wss 3792  ran crn 5358   “ cima 5360   Fn wfn 6132  ⟶wf 6133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4889  df-opab 4951  df-xp 5363  df-rel 5364  df-cnv 5365  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-fun 6139  df-fn 6140  df-f 6141

This theorem is referenced by:  k0004lem2  39416