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Mirrors > Home > MPE Home > Th. List > mapdom1 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
mapdom1 | ⊢ (𝐴 ≼ 𝐵 → (𝐴 ↑m 𝐶) ≼ (𝐵 ↑m 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldom 8945 | . . . . . . 7 ⊢ Rel ≼ | |
2 | 1 | brrelex2i 5734 | . . . . . 6 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V) |
3 | domeng 8958 | . . . . . 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 482 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) → ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)) |
7 | simpl 484 | . . . . 5 ⊢ ((𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝐴 ≈ 𝑥) | |
8 | enrefg 8980 | . . . . . 6 ⊢ (𝐶 ∈ V → 𝐶 ≈ 𝐶) | |
9 | 8 | adantl 483 | . . . . 5 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) → 𝐶 ≈ 𝐶) |
10 | mapen 9141 | . . . . 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 7442 | . . . . 5 ⊢ (𝐵 ↑m 𝐶) ∈ V | |
13 | 2 | ad2antrr 725 | . . . . . 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 8883 | . . . . . 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 8996 | . . . . 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 9008 | . . . 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 1939 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) → (𝐴 ↑m 𝐶) ≼ (𝐵 ↑m 𝐶)) |
22 | elmapex 8842 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ↑m 𝐶) → (𝐴 ∈ V ∧ 𝐶 ∈ V)) | |
23 | 22 | simprd 497 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴 ↑m 𝐶) → 𝐶 ∈ V) |
24 | 23 | con3i 154 | . . . . 5 ⊢ (¬ 𝐶 ∈ V → ¬ 𝑥 ∈ (𝐴 ↑m 𝐶)) |
25 | 24 | eq0rdv 4405 | . . . 4 ⊢ (¬ 𝐶 ∈ V → (𝐴 ↑m 𝐶) = ∅) |
26 | 25 | adantl 483 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐶 ∈ V) → (𝐴 ↑m 𝐶) = ∅) |
27 | 12 | 0dom 9106 | . . 3 ⊢ ∅ ≼ (𝐵 ↑m 𝐶) |
28 | 26, 27 | eqbrtrdi 5188 | . 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 205 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 Vcvv 3475 ⊆ wss 3949 ∅c0 4323 class class class wbr 5149 (class class class)co 7409 ↑m cmap 8820 ≈ cen 8936 ≼ cdom 8937 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-map 8822 df-en 8940 df-dom 8941 |
This theorem is referenced by: mappwen 10107 pwcfsdom 10578 cfpwsdom 10579 rpnnen 16170 rexpen 16171 hauspwdom 23005 |
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