mapfienlem1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > mapfienlem1		Structured version Visualization version GIF version

Theorem mapfienlem1 9397

Description: Lemma 1 for mapfien 9400. (Contributed by AV, 3-Jul-2019.) (Revised by AV, 28-Jul-2024.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
mapfienlem1	⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝐺 ∘ (𝑓 ∘ 𝐹)) finSupp 𝑊)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐶 𝑥,𝑓,𝐹 𝑓,𝐺,𝑥 𝜑,𝑓 𝑥,𝐷 𝑆,𝑓 𝑇,𝑓 𝑥,𝑊 𝑥,𝑍 Allowed substitution hints: 𝜑(𝑥) 𝐴(𝑓) 𝐵(𝑓) 𝐶(𝑓) 𝐷(𝑓) 𝑆(𝑥) 𝑇(𝑥) 𝑈(𝑥, 𝑓) 𝑉(𝑥, 𝑓) 𝑊(𝑓) 𝑋(𝑥, 𝑓) 𝑌(𝑥, 𝑓) 𝑍(𝑓)

**Proof of Theorem mapfienlem1**
Step	Hyp	Ref	Expression
1		mapfien.w	. . . 4 ⊢ 𝑊 = (𝐺‘𝑍)
2	1	fvexi 6899	. . 3 ⊢ 𝑊 ∈ V
3	2	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑊 ∈ V)
4		mapfien.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
5	4	adantr 486	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑍 ∈ 𝐵)
6		elrabi 3641	. . . . 5 ⊢ (𝑓 ∈ {𝑥 ∈ (𝐵 ↑_m 𝐴) ∣ 𝑥 finSupp 𝑍} → 𝑓 ∈ (𝐵 ↑_m 𝐴))
7		elmapi 8869	. . . . 5 ⊢ (𝑓 ∈ (𝐵 ↑_m 𝐴) → 𝑓:𝐴⟶𝐵)
8	6, 7	syl 18	. . . 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 2879	. . 3 ⊢ (𝑓 ∈ 𝑆 → 𝑓:𝐴⟶𝐵)
11		mapfien.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)
12		f1of 6824	. . . 4 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → 𝐹:𝐶⟶𝐴)
13	11, 12	syl 18	. . 3 ⊢ (𝜑 → 𝐹:𝐶⟶𝐴)
14		fco 6734	. . 3 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝐹:𝐶⟶𝐴) → (𝑓 ∘ 𝐹):𝐶⟶𝐵)
15	10, 13, 14	syl2anr 609	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝑓 ∘ 𝐹):𝐶⟶𝐵)
16		mapfien.g	. . . 4 ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)
17		f1of 6824	. . . 4 ⊢ (𝐺:𝐵–1-1-onto→𝐷 → 𝐺:𝐵⟶𝐷)
18	16, 17	syl 18	. . 3 ⊢ (𝜑 → 𝐺:𝐵⟶𝐷)
19	18	adantr 486	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐺:𝐵⟶𝐷)
20		ssidd 3954	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐵 ⊆ 𝐵)
21		mapfien.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
22	21	adantr 486	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐶 ∈ 𝑋)
23		mapfien.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
24	23	adantr 486	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐵 ∈ 𝑉)
25		breq1 5106	. . . . . 6 ⊢ (𝑥 = 𝑓 → (𝑥 finSupp 𝑍 ↔ 𝑓 finSupp 𝑍))
26	25, 9	elrab2 3649	. . . . 5 ⊢ (𝑓 ∈ 𝑆 ↔ (𝑓 ∈ (𝐵 ↑_m 𝐴) ∧ 𝑓 finSupp 𝑍))
27	26	simprbi 503	. . . 4 ⊢ (𝑓 ∈ 𝑆 → 𝑓 finSupp 𝑍)
28	27	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑓 finSupp 𝑍)
29		f1of1 6823	. . . . 5 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → 𝐹:𝐶–1-1→𝐴)
30	11, 29	syl 18	. . . 4 ⊢ (𝜑 → 𝐹:𝐶–1-1→𝐴)
31	30	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐹:𝐶–1-1→𝐴)
32		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑓 ∈ 𝑆)
33	28, 31, 5, 32	fsuppco 9394	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝑓 ∘ 𝐹) finSupp 𝑍)
34	1	eqcomi 2770	. . 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 9396	1 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝐺 ∘ (𝑓 ∘ 𝐹)) finSupp 𝑊)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 {crab 3413 Vcvv 3451 class class class wbr 5103 ∘ ccom 5655 ⟶wf 6534 –1-1→wf1 6535 –1-1-onto→wf1o 6537 ‘cfv 6538 (class class class)co 7420 ↑_m cmap 8847 finSupp cfsupp 9353

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-1o 8476 df-map 8849 df-en 8974 df-dom 8975 df-fin 8977 df-fsupp 9354

This theorem is used by: mapfien 9400

W3C validator