Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mapfienlem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for mapfien 8865. (Contributed by AV, 3-Jul-2019.) |
Ref | Expression |
---|---|
mapfien.s | ⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} |
mapfien.t | ⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊} |
mapfien.w | ⊢ 𝑊 = (𝐺‘𝑍) |
mapfien.f | ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴) |
mapfien.g | ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷) |
mapfien.a | ⊢ (𝜑 → 𝐴 ∈ V) |
mapfien.b | ⊢ (𝜑 → 𝐵 ∈ V) |
mapfien.c | ⊢ (𝜑 → 𝐶 ∈ V) |
mapfien.d | ⊢ (𝜑 → 𝐷 ∈ V) |
mapfien.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
Ref | Expression |
---|---|
mapfienlem3 | ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapfien.g | . . . . . . 7 ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷) | |
2 | f1ocnv 6622 | . . . . . . 7 ⊢ (𝐺:𝐵–1-1-onto→𝐷 → ◡𝐺:𝐷–1-1-onto→𝐵) | |
3 | f1of 6610 | . . . . . . 7 ⊢ (◡𝐺:𝐷–1-1-onto→𝐵 → ◡𝐺:𝐷⟶𝐵) | |
4 | 1, 2, 3 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → ◡𝐺:𝐷⟶𝐵) |
5 | 4 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ◡𝐺:𝐷⟶𝐵) |
6 | elrabi 3675 | . . . . . . . 8 ⊢ (𝑔 ∈ {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊} → 𝑔 ∈ (𝐷 ↑m 𝐶)) | |
7 | mapfien.t | . . . . . . . 8 ⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊} | |
8 | 6, 7 | eleq2s 2931 | . . . . . . 7 ⊢ (𝑔 ∈ 𝑇 → 𝑔 ∈ (𝐷 ↑m 𝐶)) |
9 | 8 | adantl 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → 𝑔 ∈ (𝐷 ↑m 𝐶)) |
10 | elmapi 8422 | . . . . . 6 ⊢ (𝑔 ∈ (𝐷 ↑m 𝐶) → 𝑔:𝐶⟶𝐷) | |
11 | 9, 10 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → 𝑔:𝐶⟶𝐷) |
12 | fco 6526 | . . . . 5 ⊢ ((◡𝐺:𝐷⟶𝐵 ∧ 𝑔:𝐶⟶𝐷) → (◡𝐺 ∘ 𝑔):𝐶⟶𝐵) | |
13 | 5, 11, 12 | syl2anc 586 | . . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → (◡𝐺 ∘ 𝑔):𝐶⟶𝐵) |
14 | mapfien.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴) | |
15 | f1ocnv 6622 | . . . . . 6 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → ◡𝐹:𝐴–1-1-onto→𝐶) | |
16 | f1of 6610 | . . . . . 6 ⊢ (◡𝐹:𝐴–1-1-onto→𝐶 → ◡𝐹:𝐴⟶𝐶) | |
17 | 14, 15, 16 | 3syl 18 | . . . . 5 ⊢ (𝜑 → ◡𝐹:𝐴⟶𝐶) |
18 | 17 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ◡𝐹:𝐴⟶𝐶) |
19 | fco 6526 | . . . 4 ⊢ (((◡𝐺 ∘ 𝑔):𝐶⟶𝐵 ∧ ◡𝐹:𝐴⟶𝐶) → ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹):𝐴⟶𝐵) | |
20 | 13, 18, 19 | syl2anc 586 | . . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹):𝐴⟶𝐵) |
21 | mapfien.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ V) | |
22 | mapfien.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ V) | |
23 | 21, 22 | elmapd 8414 | . . . 4 ⊢ (𝜑 → (((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) ∈ (𝐵 ↑m 𝐴) ↔ ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹):𝐴⟶𝐵)) |
24 | 23 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → (((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) ∈ (𝐵 ↑m 𝐴) ↔ ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹):𝐴⟶𝐵)) |
25 | 20, 24 | mpbird 259 | . 2 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) ∈ (𝐵 ↑m 𝐴)) |
26 | mapfien.s | . . 3 ⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} | |
27 | mapfien.w | . . 3 ⊢ 𝑊 = (𝐺‘𝑍) | |
28 | mapfien.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ V) | |
29 | mapfien.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ V) | |
30 | mapfien.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
31 | 26, 7, 27, 14, 1, 22, 21, 28, 29, 30 | mapfienlem2 8863 | . 2 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) finSupp 𝑍) |
32 | breq1 5062 | . . 3 ⊢ (𝑥 = ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) → (𝑥 finSupp 𝑍 ↔ ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) finSupp 𝑍)) | |
33 | 32, 26 | elrab2 3683 | . 2 ⊢ (((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) ∈ 𝑆 ↔ (((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) ∈ (𝐵 ↑m 𝐴) ∧ ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) finSupp 𝑍)) |
34 | 25, 31, 33 | sylanbrc 585 | 1 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {crab 3142 Vcvv 3495 class class class wbr 5059 ◡ccnv 5549 ∘ ccom 5554 ⟶wf 6346 –1-1-onto→wf1o 6349 ‘cfv 6350 (class class class)co 7150 ↑m cmap 8400 finSupp cfsupp 8827 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-1o 8096 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-fin 8507 df-fsupp 8828 |
This theorem is referenced by: mapfien 8865 |
Copyright terms: Public domain | W3C validator |