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Theorem mapfienlem1 9094

Description: Lemma 1 for mapfien 9097. (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 6770	. . 3 ⊢ 𝑊 ∈ V
3	2	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑊 ∈ V)
4		mapfien.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
5	4	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑍 ∈ 𝐵)
6		elrabi 3611	. . . . 5 ⊢ (𝑓 ∈ {𝑥 ∈ (𝐵 ↑_m 𝐴) ∣ 𝑥 finSupp 𝑍} → 𝑓 ∈ (𝐵 ↑_m 𝐴))
7		elmapi 8595	. . . . 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 6700	. . . 4 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → 𝐹:𝐶⟶𝐴)
13	11, 12	syl 17	. . 3 ⊢ (𝜑 → 𝐹:𝐶⟶𝐴)
14		fco 6608	. . 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 6700	. . . 4 ⊢ (𝐺:𝐵–1-1-onto→𝐷 → 𝐺:𝐵⟶𝐷)
18	16, 17	syl 17	. . 3 ⊢ (𝜑 → 𝐺:𝐵⟶𝐷)
19	18	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐺:𝐵⟶𝐷)
20		ssidd 3940	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐵 ⊆ 𝐵)
21		mapfien.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
22	21	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐶 ∈ 𝑋)
23		mapfien.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
24	23	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐵 ∈ 𝑉)
25		breq1 5073	. . . . . 6 ⊢ (𝑥 = 𝑓 → (𝑥 finSupp 𝑍 ↔ 𝑓 finSupp 𝑍))
26	25, 9	elrab2 3620	. . . . 5 ⊢ (𝑓 ∈ 𝑆 ↔ (𝑓 ∈ (𝐵 ↑_m 𝐴) ∧ 𝑓 finSupp 𝑍))
27	26	simprbi 496	. . . 4 ⊢ (𝑓 ∈ 𝑆 → 𝑓 finSupp 𝑍)
28	27	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑓 finSupp 𝑍)
29		f1of1 6699	. . . . 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 480	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐹:𝐶–1-1→𝐴)
32		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑓 ∈ 𝑆)
33	28, 31, 5, 32	fsuppco 9091	. 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 9093	1 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝐺 ∘ (𝑓 ∘ 𝐹)) finSupp 𝑊)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 {crab 3067 Vcvv 3422 class class class wbr 5070 ∘ ccom 5584 ⟶wf 6414 –1-1→wf1 6415 –1-1-onto→wf1o 6417 ‘cfv 6418 (class class class)co 7255 ↑_m cmap 8573 finSupp cfsupp 9058

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-1o 8267 df-map 8575 df-en 8692 df-fin 8695 df-fsupp 9059

This theorem is referenced by: mapfien 9097

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