mapdom1 - Metamath Proof Explorer

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Theorem mapdom1 9137

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 8940	. . . . . . 7 ⊢ Rel ≼
2	1	brrelex2i 5723	. . . . . 6 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V)
3		domeng 8953	. . . . . 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 8975	. . . . . 6 ⊢ (𝐶 ∈ V → 𝐶 ≈ 𝐶)
9	8	adantl 481	. . . . 5 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) → 𝐶 ≈ 𝐶)
10		mapen 9136	. . . . 5 ⊢ ((𝐴 ≈ 𝑥 ∧ 𝐶 ≈ 𝐶) → (𝐴 ↑_m 𝐶) ≈ (𝑥 ↑_m 𝐶))
11	7, 9, 10	syl2anr 596	. . . 4 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) ∧ (𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)) → (𝐴 ↑_m 𝐶) ≈ (𝑥 ↑_m 𝐶))
12		ovex 7434	. . . . 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 8878	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ 𝑥 ⊆ 𝐵) → (𝑥 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
16	13, 14, 15	syl2anc 583	. . . . 5 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) ∧ (𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)) → (𝑥 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
17		ssdomg 8991	. . . . 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 9003	. . . 4 ⊢ (((𝐴 ↑_m 𝐶) ≈ (𝑥 ↑_m 𝐶) ∧ (𝑥 ↑_m 𝐶) ≼ (𝐵 ↑_m 𝐶)) → (𝐴 ↑_m 𝐶) ≼ (𝐵 ↑_m 𝐶))
20	11, 18, 19	syl2anc 583	. . 3 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) ∧ (𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)) → (𝐴 ↑_m 𝐶) ≼ (𝐵 ↑_m 𝐶))
21	6, 20	exlimddv 1930	. 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) → (𝐴 ↑_m 𝐶) ≼ (𝐵 ↑_m 𝐶))
22		elmapex 8837	. . . . . . 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 4396	. . . 4 ⊢ (¬ 𝐶 ∈ V → (𝐴 ↑_m 𝐶) = ∅)
26	25	adantl 481	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐶 ∈ V) → (𝐴 ↑_m 𝐶) = ∅)
27	12	0dom 9101	. . 3 ⊢ ∅ ≼ (𝐵 ↑_m 𝐶)
28	26, 27	eqbrtrdi 5177	. 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 395 = wceq 1533 ∃wex 1773 ∈ wcel 2098 Vcvv 3466 ⊆ wss 3940 ∅c0 4314 class class class wbr 5138 (class class class)co 7401 ↑_m cmap 8815 ≈ cen 8931 ≼ cdom 8932

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-map 8817 df-en 8935 df-dom 8936

This theorem is referenced by: mappwen 10102 pwcfsdom 10573 cfpwsdom 10574 rpnnen 16166 rexpen 16167 hauspwdom 23315

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