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Theorem mapfienlem1 9164

Description: Lemma 1 for mapfien 9167. (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 6788	. . 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 3618	. . . . 5 ⊢ (𝑓 ∈ {𝑥 ∈ (𝐵 ↑_m 𝐴) ∣ 𝑥 finSupp 𝑍} → 𝑓 ∈ (𝐵 ↑_m 𝐴))
7		elmapi 8637	. . . . 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 2857	. . 3 ⊢ (𝑓 ∈ 𝑆 → 𝑓:𝐴⟶𝐵)
11		mapfien.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)
12		f1of 6716	. . . 4 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → 𝐹:𝐶⟶𝐴)
13	11, 12	syl 17	. . 3 ⊢ (𝜑 → 𝐹:𝐶⟶𝐴)
14		fco 6624	. . 3 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝐹:𝐶⟶𝐴) → (𝑓 ∘ 𝐹):𝐶⟶𝐵)
15	10, 13, 14	syl2anr 597	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝑓 ∘ 𝐹):𝐶⟶𝐵)
16		mapfien.g	. . . 4 ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)
17		f1of 6716	. . . 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 3944	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐵 ⊆ 𝐵)
21		mapfien.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
22	21	adantr 481	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐶 ∈ 𝑋)
23		mapfien.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
24	23	adantr 481	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐵 ∈ 𝑉)
25		breq1 5077	. . . . . 6 ⊢ (𝑥 = 𝑓 → (𝑥 finSupp 𝑍 ↔ 𝑓 finSupp 𝑍))
26	25, 9	elrab2 3627	. . . . 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 6715	. . . . 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 9161	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝑓 ∘ 𝐹) finSupp 𝑍)
34	1	eqcomi 2747	. . 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 9163	1 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝐺 ∘ (𝑓 ∘ 𝐹)) finSupp 𝑊)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 {crab 3068 Vcvv 3432 class class class wbr 5074 ∘ ccom 5593 ⟶wf 6429 –1-1→wf1 6430 –1-1-onto→wf1o 6432 ‘cfv 6433 (class class class)co 7275 ↑_m cmap 8615 finSupp cfsupp 9128

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-1o 8297 df-map 8617 df-en 8734 df-fin 8737 df-fsupp 9129

This theorem is referenced by: mapfien 9167

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