mapdom1 - Metamath Proof Explorer

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Theorem mapdom1 8929

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 8739	. . . . . . 7 ⊢ Rel ≼
2	1	brrelex2i 5644	. . . . . 6 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V)
3		domeng 8752	. . . . . 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 266	. . . 4 ⊢ (𝐴 ≼ 𝐵 → ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵))
6	5	adantr 481	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) → ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵))
7		simpl 483	. . . . 5 ⊢ ((𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝐴 ≈ 𝑥)
8		enrefg 8772	. . . . . 6 ⊢ (𝐶 ∈ V → 𝐶 ≈ 𝐶)
9	8	adantl 482	. . . . 5 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) → 𝐶 ≈ 𝐶)
10		mapen 8928	. . . . 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 7308	. . . . 5 ⊢ (𝐵 ↑_m 𝐶) ∈ V
13	2	ad2antrr 723	. . . . . 6 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) ∧ (𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)) → 𝐵 ∈ V)
14		simprr 770	. . . . . 6 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) ∧ (𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)) → 𝑥 ⊆ 𝐵)
15		mapss 8677	. . . . . 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 8786	. . . . 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 8798	. . . 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 1938	. 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) → (𝐴 ↑_m 𝐶) ≼ (𝐵 ↑_m 𝐶))
22		elmapex 8636	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ↑_m 𝐶) → (𝐴 ∈ V ∧ 𝐶 ∈ V))
23	22	simprd 496	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ↑_m 𝐶) → 𝐶 ∈ V)
24	23	con3i 154	. . . . 5 ⊢ (¬ 𝐶 ∈ V → ¬ 𝑥 ∈ (𝐴 ↑_m 𝐶))
25	24	eq0rdv 4338	. . . 4 ⊢ (¬ 𝐶 ∈ V → (𝐴 ↑_m 𝐶) = ∅)
26	25	adantl 482	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐶 ∈ V) → (𝐴 ↑_m 𝐶) = ∅)
27	12	0dom 8893	. . 3 ⊢ ∅ ≼ (𝐵 ↑_m 𝐶)
28	26, 27	eqbrtrdi 5113	. 2 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐶 ∈ V) → (𝐴 ↑_m 𝐶) ≼ (𝐵 ↑_m 𝐶))
29	21, 28	pm2.61dan 810	1 ⊢ (𝐴 ≼ 𝐵 → (𝐴 ↑_m 𝐶) ≼ (𝐵 ↑_m 𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∃wex 1782 ∈ wcel 2106 Vcvv 3432 ⊆ wss 3887 ∅c0 4256 class class class wbr 5074 (class class class)co 7275 ↑_m cmap 8615 ≈ cen 8730 ≼ cdom 8731

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-map 8617 df-en 8734 df-dom 8735

This theorem is referenced by: mappwen 9868 pwcfsdom 10339 cfpwsdom 10340 rpnnen 15936 rexpen 15937 hauspwdom 22652

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