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Mirrors > Home > MPE Home > Th. List > mapfienlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for mapfien 9441. (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 6916 | . . 3 ⊢ 𝑊 ∈ V |
3 | 2 | a1i 11 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑊 ∈ V) |
4 | mapfien.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
5 | 4 | adantr 479 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑍 ∈ 𝐵) |
6 | elrabi 3678 | . . . . 5 ⊢ (𝑓 ∈ {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} → 𝑓 ∈ (𝐵 ↑m 𝐴)) | |
7 | elmapi 8876 | . . . . 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 2847 | . . 3 ⊢ (𝑓 ∈ 𝑆 → 𝑓:𝐴⟶𝐵) |
11 | mapfien.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴) | |
12 | f1of 6844 | . . . 4 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → 𝐹:𝐶⟶𝐴) | |
13 | 11, 12 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹:𝐶⟶𝐴) |
14 | fco 6752 | . . 3 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝐹:𝐶⟶𝐴) → (𝑓 ∘ 𝐹):𝐶⟶𝐵) | |
15 | 10, 13, 14 | syl2anr 595 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝑓 ∘ 𝐹):𝐶⟶𝐵) |
16 | mapfien.g | . . . 4 ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷) | |
17 | f1of 6844 | . . . 4 ⊢ (𝐺:𝐵–1-1-onto→𝐷 → 𝐺:𝐵⟶𝐷) | |
18 | 16, 17 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺:𝐵⟶𝐷) |
19 | 18 | adantr 479 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐺:𝐵⟶𝐷) |
20 | ssidd 4005 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐵 ⊆ 𝐵) | |
21 | mapfien.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
22 | 21 | adantr 479 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐶 ∈ 𝑋) |
23 | mapfien.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
24 | 23 | adantr 479 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐵 ∈ 𝑉) |
25 | breq1 5155 | . . . . . 6 ⊢ (𝑥 = 𝑓 → (𝑥 finSupp 𝑍 ↔ 𝑓 finSupp 𝑍)) | |
26 | 25, 9 | elrab2 3687 | . . . . 5 ⊢ (𝑓 ∈ 𝑆 ↔ (𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑓 finSupp 𝑍)) |
27 | 26 | simprbi 495 | . . . 4 ⊢ (𝑓 ∈ 𝑆 → 𝑓 finSupp 𝑍) |
28 | 27 | adantl 480 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑓 finSupp 𝑍) |
29 | f1of1 6843 | . . . . 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 479 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐹:𝐶–1-1→𝐴) |
32 | simpr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑓 ∈ 𝑆) | |
33 | 28, 31, 5, 32 | fsuppco 9435 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝑓 ∘ 𝐹) finSupp 𝑍) |
34 | 1 | eqcomi 2737 | . . 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 9437 | 1 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝐺 ∘ (𝑓 ∘ 𝐹)) finSupp 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 {crab 3430 Vcvv 3473 class class class wbr 5152 ∘ ccom 5686 ⟶wf 6549 –1-1→wf1 6550 –1-1-onto→wf1o 6552 ‘cfv 6553 (class class class)co 7426 ↑m cmap 8853 finSupp cfsupp 9395 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-supp 8174 df-1o 8495 df-map 8855 df-en 8973 df-fin 8976 df-fsupp 9396 |
This theorem is referenced by: mapfien 9441 |
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