k0004lem2 - Mathbox for Richard Penner

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Theorem k0004lem2 44130

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 1138	. . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ⊆ 𝐵)
2		sseqin2 4188	. . . . 5 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐶) = 𝐶)
3	2	biimpi 216	. . . 4 ⊢ (𝐶 ⊆ 𝐵 → (𝐵 ∩ 𝐶) = 𝐶)
4	3	eqcomd 2736	. . 3 ⊢ (𝐶 ⊆ 𝐵 → 𝐶 = (𝐵 ∩ 𝐶))
5		k0004lem1 44129	. . 3 ⊢ (𝐶 = (𝐵 ∩ 𝐶) → ((𝐹:𝐴⟶𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ 𝐹:𝐴⟶𝐶))
6	1, 4, 5	3syl 18	. 2 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵) → ((𝐹:𝐴⟶𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ 𝐹:𝐴⟶𝐶))
7		simp2 1137	. . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵) → 𝐵 ∈ 𝑉)
8		simp1 1136	. . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵) → 𝐴 ∈ 𝑈)
9	7, 8	elmapd 8815	. . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵) → (𝐹 ∈ (𝐵 ↑_m 𝐴) ↔ 𝐹:𝐴⟶𝐵))
10	9	anbi1d 631	. 2 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵) → ((𝐹 ∈ (𝐵 ↑_m 𝐴) ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ (𝐹:𝐴⟶𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶)))
11	7, 1	ssexd 5281	. . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ∈ V)
12	11, 8	elmapd 8815	. 2 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵) → (𝐹 ∈ (𝐶 ↑_m 𝐴) ↔ 𝐹:𝐴⟶𝐶))
13	6, 10, 12	3bitr4d 311	1 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵) → ((𝐹 ∈ (𝐵 ↑_m 𝐴) ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ 𝐹 ∈ (𝐶 ↑_m 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 Vcvv 3450 ∩ cin 3915 ⊆ wss 3916 “ cima 5643 ⟶wf 6509 (class class class)co 7389 ↑_m cmap 8801

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3756 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-br 5110 df-opab 5172 df-id 5535 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-fv 6521 df-ov 7392 df-oprab 7393 df-mpo 7394 df-map 8803

This theorem is referenced by: k0004lem3 44131

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