![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mapfienlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for mapfien 9352. (Contributed by AV, 3-Jul-2019.) (Revised by AV, 28-Jul-2024.) |
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 ∈ 𝑈) |
mapfien.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
mapfien.c | ⊢ (𝜑 → 𝐶 ∈ 𝑋) |
mapfien.d | ⊢ (𝜑 → 𝐷 ∈ 𝑌) |
mapfien.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
Ref | Expression |
---|---|
mapfienlem1 | ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝐺 ∘ (𝑓 ∘ 𝐹)) finSupp 𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapfien.w | . . . 4 ⊢ 𝑊 = (𝐺‘𝑍) | |
2 | 1 | fvexi 6860 | . . 3 ⊢ 𝑊 ∈ V |
3 | 2 | a1i 11 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑊 ∈ V) |
4 | mapfien.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
5 | 4 | adantr 482 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑍 ∈ 𝐵) |
6 | elrabi 3643 | . . . . 5 ⊢ (𝑓 ∈ {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} → 𝑓 ∈ (𝐵 ↑m 𝐴)) | |
7 | elmapi 8793 | . . . . 5 ⊢ (𝑓 ∈ (𝐵 ↑m 𝐴) → 𝑓:𝐴⟶𝐵) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝑓 ∈ {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} → 𝑓:𝐴⟶𝐵) |
9 | mapfien.s | . . . 4 ⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} | |
10 | 8, 9 | eleq2s 2852 | . . 3 ⊢ (𝑓 ∈ 𝑆 → 𝑓:𝐴⟶𝐵) |
11 | mapfien.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴) | |
12 | f1of 6788 | . . . 4 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → 𝐹:𝐶⟶𝐴) | |
13 | 11, 12 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹:𝐶⟶𝐴) |
14 | fco 6696 | . . 3 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝐹:𝐶⟶𝐴) → (𝑓 ∘ 𝐹):𝐶⟶𝐵) | |
15 | 10, 13, 14 | syl2anr 598 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝑓 ∘ 𝐹):𝐶⟶𝐵) |
16 | mapfien.g | . . . 4 ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷) | |
17 | f1of 6788 | . . . 4 ⊢ (𝐺:𝐵–1-1-onto→𝐷 → 𝐺:𝐵⟶𝐷) | |
18 | 16, 17 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺:𝐵⟶𝐷) |
19 | 18 | adantr 482 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐺:𝐵⟶𝐷) |
20 | ssidd 3971 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐵 ⊆ 𝐵) | |
21 | mapfien.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
22 | 21 | adantr 482 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐶 ∈ 𝑋) |
23 | mapfien.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
24 | 23 | adantr 482 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐵 ∈ 𝑉) |
25 | breq1 5112 | . . . . . 6 ⊢ (𝑥 = 𝑓 → (𝑥 finSupp 𝑍 ↔ 𝑓 finSupp 𝑍)) | |
26 | 25, 9 | elrab2 3652 | . . . . 5 ⊢ (𝑓 ∈ 𝑆 ↔ (𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑓 finSupp 𝑍)) |
27 | 26 | simprbi 498 | . . . 4 ⊢ (𝑓 ∈ 𝑆 → 𝑓 finSupp 𝑍) |
28 | 27 | adantl 483 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑓 finSupp 𝑍) |
29 | f1of1 6787 | . . . . 5 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → 𝐹:𝐶–1-1→𝐴) | |
30 | 11, 29 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:𝐶–1-1→𝐴) |
31 | 30 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐹:𝐶–1-1→𝐴) |
32 | simpr 486 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑓 ∈ 𝑆) | |
33 | 28, 31, 5, 32 | fsuppco 9346 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝑓 ∘ 𝐹) finSupp 𝑍) |
34 | 1 | eqcomi 2742 | . . 3 ⊢ (𝐺‘𝑍) = 𝑊 |
35 | 34 | a1i 11 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝐺‘𝑍) = 𝑊) |
36 | 3, 5, 15, 19, 20, 22, 24, 33, 35 | fsuppcor 9348 | 1 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝐺 ∘ (𝑓 ∘ 𝐹)) finSupp 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {crab 3406 Vcvv 3447 class class class wbr 5109 ∘ ccom 5641 ⟶wf 6496 –1-1→wf1 6497 –1-1-onto→wf1o 6499 ‘cfv 6500 (class class class)co 7361 ↑m cmap 8771 finSupp cfsupp 9311 |
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-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-1o 8416 df-map 8773 df-en 8890 df-fin 8893 df-fsupp 9312 |
This theorem is referenced by: mapfien 9352 |
Copyright terms: Public domain | W3C validator |