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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 8990 | . . . . . . 7 ⊢ Rel ≼ | |
2 | 1 | brrelex2i 5746 | . . . . . 6 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V) |
3 | domeng 9002 | . . . . . 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 9023 | . . . . . 6 ⊢ (𝐶 ∈ V → 𝐶 ≈ 𝐶) | |
9 | 8 | adantl 481 | . . . . 5 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) → 𝐶 ≈ 𝐶) |
10 | mapen 9180 | . . . . 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 7464 | . . . . 5 ⊢ (𝐵 ↑m 𝐶) ∈ V | |
13 | 2 | ad2antrr 726 | . . . . . 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 8928 | . . . . . 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 9039 | . . . . 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 9051 | . . . 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 1933 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) → (𝐴 ↑m 𝐶) ≼ (𝐵 ↑m 𝐶)) |
22 | elmapex 8887 | . . . . . . 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 4413 | . . . 4 ⊢ (¬ 𝐶 ∈ V → (𝐴 ↑m 𝐶) = ∅) |
26 | 25 | adantl 481 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐶 ∈ V) → (𝐴 ↑m 𝐶) = ∅) |
27 | 12 | 0dom 9145 | . . 3 ⊢ ∅ ≼ (𝐵 ↑m 𝐶) |
28 | 26, 27 | eqbrtrdi 5187 | . 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 1537 ∃wex 1776 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 ∅c0 4339 class class class wbr 5148 (class class class)co 7431 ↑m cmap 8865 ≈ cen 8981 ≼ cdom 8982 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-map 8867 df-en 8985 df-dom 8986 |
This theorem is referenced by: mappwen 10150 pwcfsdom 10621 cfpwsdom 10622 rpnnen 16260 rexpen 16261 hauspwdom 23525 |
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