mapdom1 - Metamath Proof Explorer

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Theorem mapdom1 9071

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 8890	. . . . . . 7 ⊢ Rel ≼
2	1	brrelex2i 5679	. . . . . 6 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V)
3		domeng 8900	. . . . . 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 8922	. . . . . 6 ⊢ (𝐶 ∈ V → 𝐶 ≈ 𝐶)
9	8	adantl 481	. . . . 5 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) → 𝐶 ≈ 𝐶)
10		mapen 9070	. . . . 5 ⊢ ((𝐴 ≈ 𝑥 ∧ 𝐶 ≈ 𝐶) → (𝐴 ↑_m 𝐶) ≈ (𝑥 ↑_m 𝐶))
11	7, 9, 10	syl2anr 598	. . . 4 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) ∧ (𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)) → (𝐴 ↑_m 𝐶) ≈ (𝑥 ↑_m 𝐶))
12		ovex 7391	. . . . 5 ⊢ (𝐵 ↑_m 𝐶) ∈ V
13	2	ad2antrr 727	. . . . . 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 8828	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ 𝑥 ⊆ 𝐵) → (𝑥 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
16	13, 14, 15	syl2anc 585	. . . . 5 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) ∧ (𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)) → (𝑥 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
17		ssdomg 8938	. . . . 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 8950	. . . 4 ⊢ (((𝐴 ↑_m 𝐶) ≈ (𝑥 ↑_m 𝐶) ∧ (𝑥 ↑_m 𝐶) ≼ (𝐵 ↑_m 𝐶)) → (𝐴 ↑_m 𝐶) ≼ (𝐵 ↑_m 𝐶))
20	11, 18, 19	syl2anc 585	. . 3 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) ∧ (𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)) → (𝐴 ↑_m 𝐶) ≼ (𝐵 ↑_m 𝐶))
21	6, 20	exlimddv 1937	. 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) → (𝐴 ↑_m 𝐶) ≼ (𝐵 ↑_m 𝐶))
22		elmapex 8786	. . . . . . 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 4348	. . . 4 ⊢ (¬ 𝐶 ∈ V → (𝐴 ↑_m 𝐶) = ∅)
26	25	adantl 481	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐶 ∈ V) → (𝐴 ↑_m 𝐶) = ∅)
27	12	0dom 9036	. . 3 ⊢ ∅ ≼ (𝐵 ↑_m 𝐶)
28	26, 27	eqbrtrdi 5125	. 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 1542 ∃wex 1781 ∈ wcel 2114 Vcvv 3430 ⊆ wss 3890 ∅c0 4274 class class class wbr 5086 (class class class)co 7358 ↑_m cmap 8764 ≈ cen 8881 ≼ cdom 8882

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7933 df-2nd 7934 df-map 8766 df-en 8885 df-dom 8886

This theorem is referenced by: mappwen 10023 pwcfsdom 10495 cfpwsdom 10496 rpnnen 16183 rexpen 16184 hauspwdom 23475

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