k0004lem2 - Mathbox for Richard Penner

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Theorem k0004lem2 45148

Description: A mapping with a particular restricted range is also a mapping to that range. (Contributed by RP, 1-Apr-2021.)

**Assertion**
Ref	Expression
k0004lem2	⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵) → ((𝐹 ∈ (𝐵 ↑_m 𝐴) ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ 𝐹 ∈ (𝐶 ↑_m 𝐴)))

**Proof of Theorem k0004lem2**
Step	Hyp	Ref	Expression
1		simp3 1156	. . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ⊆ 𝐵)
2		sseqin2 4120	. . . . 5 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐶) = 𝐶)
3	2	biimpi 219	. . . 4 ⊢ (𝐶 ⊆ 𝐵 → (𝐵 ∩ 𝐶) = 𝐶)
4	3	eqcomd 2766	. . 3 ⊢ (𝐶 ⊆ 𝐵 → 𝐶 = (𝐵 ∩ 𝐶))
5		k0004lem1 45147	. . 3 ⊢ (𝐶 = (𝐵 ∩ 𝐶) → ((𝐹:𝐴⟶𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ 𝐹:𝐴⟶𝐶))
6	1, 4, 5	3syl 19	. 2 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵) → ((𝐹:𝐴⟶𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ 𝐹:𝐴⟶𝐶))
7		simp2 1155	. . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵) → 𝐵 ∈ 𝑉)
8		simp1 1154	. . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵) → 𝐴 ∈ 𝑈)
9	7, 8	elmapd 8860	. . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵) → (𝐹 ∈ (𝐵 ↑_m 𝐴) ↔ 𝐹:𝐴⟶𝐵))
10	9	anbi1d 643	. 2 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵) → ((𝐹 ∈ (𝐵 ↑_m 𝐴) ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ (𝐹:𝐴⟶𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶)))
11	7, 1	ssexd 5285	. . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ∈ V)
12	11, 8	elmapd 8860	. 2 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵) → (𝐹 ∈ (𝐶 ↑_m 𝐴) ↔ 𝐹:𝐴⟶𝐶))
13	6, 10, 12	3bitr4d 314	1 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵) → ((𝐹 ∈ (𝐵 ↑_m 𝐴) ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ 𝐹 ∈ (𝐶 ↑_m 𝐴)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 Vcvv 3450 ⊆ wss 3895 ∩ cin 4089 “ cima 5653 ⟶wf 6534 (class class class)co 7420 ↑_m cmap 8847

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-pow 5326 ax-pr 5390 ax-un 7751

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3739 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-br 5103 df-opab 5167 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-fv 6546 df-ov 7423 df-oprab 7424 df-mpo 7425 df-map 8849

This theorem is used by: k0004lem3 45149

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