mapdom1 - Metamath Proof Explorer

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Theorem mapdom1 9055

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 8875	. . . . . . 7 ⊢ Rel ≼
2	1	brrelex2i 5671	. . . . . 6 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V)
3		domeng 8885	. . . . . 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 8906	. . . . . 6 ⊢ (𝐶 ∈ V → 𝐶 ≈ 𝐶)
9	8	adantl 481	. . . . 5 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) → 𝐶 ≈ 𝐶)
10		mapen 9054	. . . . 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 7379	. . . . 5 ⊢ (𝐵 ↑_m 𝐶) ∈ V
13	2	ad2antrr 726	. . . . . 6 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) ∧ (𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)) → 𝐵 ∈ V)
14		simprr 772	. . . . . 6 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) ∧ (𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)) → 𝑥 ⊆ 𝐵)
15		mapss 8813	. . . . . 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 8922	. . . . 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 8934	. . . 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 1936	. 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) → (𝐴 ↑_m 𝐶) ≼ (𝐵 ↑_m 𝐶))
22		elmapex 8772	. . . . . . 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 4354	. . . 4 ⊢ (¬ 𝐶 ∈ V → (𝐴 ↑_m 𝐶) = ∅)
26	25	adantl 481	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐶 ∈ V) → (𝐴 ↑_m 𝐶) = ∅)
27	12	0dom 9020	. . 3 ⊢ ∅ ≼ (𝐵 ↑_m 𝐶)
28	26, 27	eqbrtrdi 5128	. 2 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐶 ∈ V) → (𝐴 ↑_m 𝐶) ≼ (𝐵 ↑_m 𝐶))
29	21, 28	pm2.61dan 812	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 1541 ∃wex 1780 ∈ wcel 2111 Vcvv 3436 ⊆ wss 3897 ∅c0 4280 class class class wbr 5089 (class class class)co 7346 ↑_m cmap 8750 ≈ cen 8866 ≼ cdom 8867

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-ov 7349 df-oprab 7350 df-mpo 7351 df-1st 7921 df-2nd 7922 df-map 8752 df-en 8870 df-dom 8871

This theorem is referenced by: mappwen 10003 pwcfsdom 10474 cfpwsdom 10475 rpnnen 16136 rexpen 16137 hauspwdom 23416

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