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Mirrors > Home > MPE Home > Th. List > mapfienlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for mapfien 8859. (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 |
---|---|
mapfienlem1 | ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝐺 ∘ (𝑓 ∘ 𝐹)) finSupp 𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapfien.w | . . . 4 ⊢ 𝑊 = (𝐺‘𝑍) | |
2 | 1 | fvexi 6677 | . . 3 ⊢ 𝑊 ∈ V |
3 | 2 | a1i 11 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑊 ∈ V) |
4 | mapfien.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
5 | 4 | adantr 481 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑍 ∈ 𝐵) |
6 | elrabi 3672 | . . . . 5 ⊢ (𝑓 ∈ {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} → 𝑓 ∈ (𝐵 ↑m 𝐴)) | |
7 | elmapi 8417 | . . . . 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 2928 | . . 3 ⊢ (𝑓 ∈ 𝑆 → 𝑓:𝐴⟶𝐵) |
11 | mapfien.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴) | |
12 | f1of 6608 | . . . 4 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → 𝐹:𝐶⟶𝐴) | |
13 | 11, 12 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹:𝐶⟶𝐴) |
14 | fco 6524 | . . 3 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝐹:𝐶⟶𝐴) → (𝑓 ∘ 𝐹):𝐶⟶𝐵) | |
15 | 10, 13, 14 | syl2anr 596 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝑓 ∘ 𝐹):𝐶⟶𝐵) |
16 | mapfien.g | . . . 4 ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷) | |
17 | f1of 6608 | . . . 4 ⊢ (𝐺:𝐵–1-1-onto→𝐷 → 𝐺:𝐵⟶𝐷) | |
18 | 16, 17 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺:𝐵⟶𝐷) |
19 | 18 | adantr 481 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐺:𝐵⟶𝐷) |
20 | ssidd 3987 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐵 ⊆ 𝐵) | |
21 | mapfien.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ V) | |
22 | 21 | adantr 481 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐶 ∈ V) |
23 | mapfien.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) | |
24 | 23 | adantr 481 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐵 ∈ V) |
25 | breq1 5060 | . . . . . 6 ⊢ (𝑥 = 𝑓 → (𝑥 finSupp 𝑍 ↔ 𝑓 finSupp 𝑍)) | |
26 | 25, 9 | elrab2 3680 | . . . . 5 ⊢ (𝑓 ∈ 𝑆 ↔ (𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑓 finSupp 𝑍)) |
27 | 26 | simprbi 497 | . . . 4 ⊢ (𝑓 ∈ 𝑆 → 𝑓 finSupp 𝑍) |
28 | 27 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑓 finSupp 𝑍) |
29 | f1of1 6607 | . . . . 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 481 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐹:𝐶–1-1→𝐴) |
32 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑓 ∈ 𝑆) | |
33 | 28, 31, 5, 32 | fsuppco 8853 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝑓 ∘ 𝐹) finSupp 𝑍) |
34 | 1 | eqcomi 2827 | . . 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 8855 | 1 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝐺 ∘ (𝑓 ∘ 𝐹)) finSupp 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {crab 3139 Vcvv 3492 class class class wbr 5057 ∘ ccom 5552 ⟶wf 6344 –1-1→wf1 6345 –1-1-onto→wf1o 6347 ‘cfv 6348 (class class class)co 7145 ↑m cmap 8395 finSupp cfsupp 8821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-1o 8091 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-fin 8501 df-fsupp 8822 |
This theorem is referenced by: mapfien 8859 |
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