mapdom1 - Metamath Proof Explorer

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Theorem mapdom1 9182

Description: Order-preserving property of set exponentiation. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)

**Assertion**
Ref	Expression
mapdom1	⊢ (𝐴 ≼ 𝐵 → (𝐴 ↑_m 𝐶) ≼ (𝐵 ↑_m 𝐶))

Proof of Theorem mapdom1 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reldom 8991	. . . . . . 7 ⊢ Rel ≼
2	1	brrelex2i 5742	. . . . . 6 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V)
3		domeng 9003	. . . . . 6 ⊢ (𝐵 ∈ V → (𝐴 ≼ 𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)))
4	2, 3	syl 17	. . . . 5 ⊢ (𝐴 ≼ 𝐵 → (𝐴 ≼ 𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)))
5	4	ibi 267	. . . 4 ⊢ (𝐴 ≼ 𝐵 → ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵))
6	5	adantr 480	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) → ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵))
7		simpl 482	. . . . 5 ⊢ ((𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝐴 ≈ 𝑥)
8		enrefg 9024	. . . . . 6 ⊢ (𝐶 ∈ V → 𝐶 ≈ 𝐶)
9	8	adantl 481	. . . . 5 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) → 𝐶 ≈ 𝐶)
10		mapen 9181	. . . . 5 ⊢ ((𝐴 ≈ 𝑥 ∧ 𝐶 ≈ 𝐶) → (𝐴 ↑_m 𝐶) ≈ (𝑥 ↑_m 𝐶))
11	7, 9, 10	syl2anr 597	. . . 4 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) ∧ (𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)) → (𝐴 ↑_m 𝐶) ≈ (𝑥 ↑_m 𝐶))
12		ovex 7464	. . . . 5 ⊢ (𝐵 ↑_m 𝐶) ∈ V
13	2	ad2antrr 726	. . . . . 6 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) ∧ (𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)) → 𝐵 ∈ V)
14		simprr 773	. . . . . 6 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) ∧ (𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)) → 𝑥 ⊆ 𝐵)
15		mapss 8929	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ 𝑥 ⊆ 𝐵) → (𝑥 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
16	13, 14, 15	syl2anc 584	. . . . 5 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) ∧ (𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)) → (𝑥 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
17		ssdomg 9040	. . . . 5 ⊢ ((𝐵 ↑_m 𝐶) ∈ V → ((𝑥 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶) → (𝑥 ↑_m 𝐶) ≼ (𝐵 ↑_m 𝐶)))
18	12, 16, 17	mpsyl 68	. . . 4 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) ∧ (𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)) → (𝑥 ↑_m 𝐶) ≼ (𝐵 ↑_m 𝐶))
19		endomtr 9052	. . . 4 ⊢ (((𝐴 ↑_m 𝐶) ≈ (𝑥 ↑_m 𝐶) ∧ (𝑥 ↑_m 𝐶) ≼ (𝐵 ↑_m 𝐶)) → (𝐴 ↑_m 𝐶) ≼ (𝐵 ↑_m 𝐶))
20	11, 18, 19	syl2anc 584	. . 3 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) ∧ (𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)) → (𝐴 ↑_m 𝐶) ≼ (𝐵 ↑_m 𝐶))
21	6, 20	exlimddv 1935	. 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) → (𝐴 ↑_m 𝐶) ≼ (𝐵 ↑_m 𝐶))
22		elmapex 8888	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ↑_m 𝐶) → (𝐴 ∈ V ∧ 𝐶 ∈ V))
23	22	simprd 495	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ↑_m 𝐶) → 𝐶 ∈ V)
24	23	con3i 154	. . . . 5 ⊢ (¬ 𝐶 ∈ V → ¬ 𝑥 ∈ (𝐴 ↑_m 𝐶))
25	24	eq0rdv 4407	. . . 4 ⊢ (¬ 𝐶 ∈ V → (𝐴 ↑_m 𝐶) = ∅)
26	25	adantl 481	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐶 ∈ V) → (𝐴 ↑_m 𝐶) = ∅)
27	12	0dom 9146	. . 3 ⊢ ∅ ≼ (𝐵 ↑_m 𝐶)
28	26, 27	eqbrtrdi 5182	. 2 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐶 ∈ V) → (𝐴 ↑_m 𝐶) ≼ (𝐵 ↑_m 𝐶))
29	21, 28	pm2.61dan 813	1 ⊢ (𝐴 ≼ 𝐵 → (𝐴 ↑_m 𝐶) ≼ (𝐵 ↑_m 𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 class class class wbr 5143 (class class class)co 7431 ↑_m cmap 8866 ≈ cen 8982 ≼ cdom 8983

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 df-en 8986 df-dom 8987

This theorem is referenced by: mappwen 10152 pwcfsdom 10623 cfpwsdom 10624 rpnnen 16263 rexpen 16264 hauspwdom 23509

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